123 HFOPDE, chapter 5.2.2

 123.1 Problem 1
 123.2 Problem 2
 123.3 Problem 3
 123.4 Problem 4
 123.5 Problem 5
 123.6 Problem 6
 123.7 Problem 7
 123.8 Problem 8

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123.1 Problem 1

problem number 997

Added March 10, 2019.

Problem Chapter 5.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c w + \beta x y+\gamma \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{-2 a b \beta +c^3 e^{\frac{c x}{a}} c_1\left (\frac{a y-b x}{a}\right )-a \beta c y-b \beta c x-\beta c^2 x y-c^2 \gamma }{c^3}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) = \left ( -{\frac{\beta \,y}{c}}-{\frac{b\beta }{{c}^{2}}} \right ) x-{\frac{\beta \,ay}{{c}^{2}}}+{\frac{1}{{c}^{3}} \left ({{\rm e}^{{\frac{cx}{a}}}}{\it \_F1} \left ({\frac{ya-bx}{a}} \right ){c}^{3}-\gamma \,{c}^{2}-2\,ab\beta \right ) } \]

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123.2 Problem 2

problem number 998

Added March 10, 2019.

Problem Chapter 5.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c w + x(\beta x+\gamma y)+\delta \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{-2 a^2 \beta +c^3 e^{\frac{c x}{a}} c_1\left (\frac{a y-b x}{a}\right )-2 a b \gamma -2 a \beta c x-a c \gamma y-b c \gamma x-\beta c^2 x^2-c^2 \delta -c^2 \gamma x y}{c^3}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =-{\frac{\beta \,{x}^{2}}{c}}+ \left ( -{\frac{\gamma \,y}{c}}+{\frac{-2\,\beta \,ac-bc\gamma }{{c}^{3}}} \right ) x-{\frac{\gamma \,ay}{{c}^{2}}}+{\frac{1}{{c}^{3}} \left ({{\rm e}^{{\frac{cx}{a}}}}{\it \_F1} \left ({\frac{ya-bx}{a}} \right ){c}^{3}-2\,ab\gamma -2\,{a}^{2}\beta -\delta \,{c}^{2} \right ) } \]

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123.3 Problem 3

problem number 999

Added March 10, 2019.

Problem Chapter 5.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + y w_y = w + a x^2+b y^2+c \]

Mathematica

\[ \left \{\left \{w(x,y)\to a x^2+b y^2+x c_1\left (\frac{y}{x}\right )-c\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =b{y}^{2}+a{x}^{2}+{\it \_F1} \left ({\frac{y}{x}} \right ) x-c \]

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123.4 Problem 4

problem number 1000

Added March 10, 2019.

Problem Chapter 5.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x w_x + b y w_y = c w + x(\beta x+\gamma y)+ \delta \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{2 a^2 c x^{\frac{c}{a}} c_1\left (y x^{-\frac{b}{a}}\right )-2 a^2 \delta +c^3 x^{\frac{c}{a}} c_1\left (y x^{-\frac{b}{a}}\right )-3 a c^2 x^{\frac{c}{a}} c_1\left (y x^{-\frac{b}{a}}\right )-b c^2 x^{\frac{c}{a}} c_1\left (y x^{-\frac{b}{a}}\right )+2 a b c x^{\frac{c}{a}} c_1\left (y x^{-\frac{b}{a}}\right )-2 a b \delta +a \beta c x^2+3 a c \delta +2 a c \gamma x y+b \beta c x^2+b c \delta -\beta c^2 x^2-c^2 \delta -c^2 \gamma x y}{c (c-2 a) (-a-b+c)}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{\gamma \,y}{a}{x}^{{\frac{a+b}{a}}-{\frac{b}{a}}} \left ({\frac{-c+b}{a}}+1 \right ) ^{-1}}+{\frac{\beta \,{x}^{2}}{a} \left ({\frac{a-c}{a}}+1 \right ) ^{-1}}+{\frac{\delta }{a} \left ( 1-{\frac{a+c}{a}} \right ) ^{-1}}+{x}^{{\frac{c}{a}}}{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ) \]

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123.5 Problem 5

problem number 1001

Added March 10, 2019.

Problem Chapter 5.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a y w_x + (b_2 x^2+b_1 x+b_0) w_y = (c_2 x^2+c_1 x+c_0) w + s_2 x^2+s_1 x+s_0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ \text{Too large to display} \]

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123.6 Problem 6

problem number 1002

Added March 10, 2019.

Problem Chapter 5.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a y^2 w_x + (b_1 x^2+b_0) w_y = (c_1 x^2+c_0) w + s_1 x^2+s_0 \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) =- \left ( \sqrt{3}\int ^{x}\!{\frac{{{\it \_b}}^{2}{\it s1}+{\it s0}}{\sqrt{a \left ( 2\,{{\it \_b}}^{3}{\it b1}-2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}+6\,{\it \_b}\,{\it b0}-6\,{\it b0}\,x \right ) }}{{\rm e}^{\sqrt{3} \left ({\frac{2/3\,i{\it c0}\,\sqrt{3}}{\sqrt{2\,{{\it \_b}}^{3}a{\it b1}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) +6\,{\it \_b}\,a{\it b0}}} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \sqrt{{-i\sqrt{3} \left ({\it \_b}+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt{{1 \left ({\it \_b}-1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt{{i\sqrt{3} \left ({\it \_b}+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt{3}\sqrt{{-i\sqrt{3} \left ({\it \_b}+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt{{-i\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) }+{\it c1}\, \left ( 1/3\,{\frac{\sqrt{2\,{{\it \_b}}^{3}a{\it b1}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) +6\,{\it \_b}\,a{\it b0}}}{a{\it b1}}}-{\frac{2/3\,i{\it b0}\,\sqrt{3}}{{\it b1}\,\sqrt{2\,{{\it \_b}}^{3}a{\it b1}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) +6\,{\it \_b}\,a{\it b0}}} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \sqrt{{-i\sqrt{3} \left ({\it \_b}+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt{{1 \left ({\it \_b}-1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt{{i\sqrt{3} \left ({\it \_b}+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt{3}\sqrt{{-i\sqrt{3} \left ({\it \_b}+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt{{-i\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) } \right ) \right ) }}}{d{\it \_b}}-{\it \_F1} \left ( 1/3\,{\frac{-2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x}{a}} \right ) \right ){{\rm e}^{\sqrt{3} \left ({\frac{-2/3\,i{\it c0}\,\sqrt{3}}{\sqrt{2\,a{\it b1}\,{x}^{3}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) +6\,a{\it b0}\,x}} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \sqrt{{-i\sqrt{3} \left ( x+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt{{1 \left ( x-1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt{{i\sqrt{3} \left ( x+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt{3}\sqrt{{-i\sqrt{3} \left ( x+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt{{-i\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) }-{\it c1}\, \left ( 1/3\,{\frac{\sqrt{2\,a{\it b1}\,{x}^{3}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) +6\,a{\it b0}\,x}}{a{\it b1}}}-{\frac{2/3\,i{\it b0}\,\sqrt{3}}{{\it b1}\,\sqrt{2\,a{\it b1}\,{x}^{3}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) +6\,a{\it b0}\,x}} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \sqrt{{-i\sqrt{3} \left ( x+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt{{1 \left ( x-1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt{{i\sqrt{3} \left ( x+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt{3}\sqrt{{-i\sqrt{3} \left ( x+1/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}-{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}+i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt{{-i\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+3\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}}-i/2\sqrt{3} \left ( 1/2\,{\frac{1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}+2\,{{\it b0}{\frac{1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,a{y}^{2}+12\,{\it b0}\,x+2\,\sqrt{{\frac{ \left ( -2\,{\it b1}\,{x}^{3}+3\,a{y}^{2}-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ){{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) } \right ) \right ) }} \]

_______________________________________________________________________________________

123.7 Problem 7

problem number 1003

Added March 10, 2019.

Problem Chapter 5.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a_1 x^2+a_0) w_x + (y+b_2 x^2+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_{22}y^2+k{12} x y+k_{11} x^2+k_0 \]

Mathematica

\[ \text{DSolve}\left [y^2 \left (\text{a0}+\text{a1} x^2\right ) w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (\text{b0}+\text{b1} x+\text{b2} x^2+y\right )=w(x,y) (\text{c0}+\text{c1} x+\text{c2} y)+\text{k0}+\text{k11} x^2+\text{k12} x y+\text{k22} y^2,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( \left ( \int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f} \right ) ^{2}{\it k22}\,{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}} \left ( \int \!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt{{\it a0}\,{\it a1}}-2\,\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) \right ) }}}+2\,\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ){\it k22}\,{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}} \left ( \int \!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt{{\it a0}\,{\it a1}}-2\,\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) \right ) }}}+ \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) ^{2}{\it k22}\,{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}} \left ( \int \!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt{{\it a0}\,{\it a1}}-2\,\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) \right ) }}}+{\it k12}\,{\it \_f}\,\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}} \left ( \int \!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt{{\it a0}\,{\it a1}}-\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) \right ) }}}+{\it k12}\,{\it \_f}\, \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ){{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}} \left ( \int \!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt{{\it a0}\,{\it a1}}-\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) \right ) }}}+{{\rm e}^{-\int \!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}}}{\it k11}\,{{\it \_f}}^{2}+{{\rm e}^{-\int \!{\frac{1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_f}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}}}{\it k0} \right ) }{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) \right ){{\rm e}^{\int ^{x}\!{\frac{1}{{{\it \_b}}^{2}{\it a1}+{\it a0}} \left ({\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_b}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac{{{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0}}{{{\it \_b}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_b}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_b}+{\it c2}\,{{\rm e}^{{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{{\it \_b}\,{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac{{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac{1}{\sqrt{{\it a0}\,{\it a1}}}\arctan \left ({\frac{x{\it a1}}{\sqrt{{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_b}+{\it c0} \right ) }{d{\it \_b}}}} \]

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123.8 Problem 8

problem number 1004

Added March 10, 2019.

Problem Chapter 5.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a_1 x^2+a_0) w_x + (b_2 y^2+b_1 x y) w_y = (c_2 y^2+c_1 x^2) w + s_{22}y^2+s_{12} x y+s_{11} x^2+s_0 \]

Mathematica

\[ \text{DSolve}\left [y^2 \left (\text{a0}+\text{a1} x^2\right ) w^{(1,0)}(x,y)+w^{(0,1)}(x,y) \left (\text{b1} x^2+\text{b2} y^2\right )=w(x,y) \left (\text{c1} x^2+\text{c2} y^2\right )+\text{s0}+\text{s11} x^2+\text{s12} x y+\text{s22} y^2,w(x,y),\{x,y\}\right ] \]

Maple

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