37 HFOPDE, chapter 2.2.3

 37.1 problem number 1
 37.2 problem number 2
 37.3 problem number 3
 37.4 problem number 4
 37.5 problem number 5
 37.6 problem number 6
 37.7 problem number 7
 37.8 problem number 8
 37.9 problem number 9

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37.1 problem number 1

problem number 254

Added January 2, 2019.

Problem 2.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(y^2+b x^2 y-a^2-a b x^2)w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{b x^3+2 x y+2}{b x^2+2 y}\right )\right \}\right \} \] But it can’t solve it when assuming \(b>0\) which is strange.

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({1 \left ( -3\,{\it csgn} \left ( b \right ){\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) b{x}^{2}+2\,{3}^{2/3}\sqrt [6]{{b}^{2}}{\it HeunTPrime} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) +3\,{\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) b{x}^{2}+6\,{\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) y \right ){\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \left ( 3\,{\it csgn} \left ( b \right ) \left ({\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{2}\int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt{{b}^{2}}}} \left ({\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}xb{x}^{2}-2\,\sqrt [6]{{b}^{2}}{3}^{2/3}{\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ){\it HeunTPrime} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt{{b}^{2}}}} \left ({\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}x-3\, \left ({\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{2}\int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt{{b}^{2}}}} \left ({\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}xb{x}^{2}-6\, \left ({\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{2}\int \!{{{\rm e}^{1/3\,{x}^{3}\sqrt{{b}^{2}}}} \left ({\it HeunT} \left ( -{\frac{{a}^{2}{3}^{2/3}}{\sqrt [3]{{b}^{2}}}},-3\,{\frac{ \left ( a-1 \right ) \sqrt{{b}^{2}}}{b}},0,1/3\,{3}^{2/3}\sqrt [6]{{b}^{2}}x \right ) \right ) ^{-2}}\,{\rm d}xy-6\,{{\rm e}^{1/3\,{x}^{3}\sqrt{{b}^{2}}}} \right ) ^{-1}} \right ) \] Mathematica solution is much simpler

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37.2 problem number 2

problem number 255

Added January 2, 2019.

Problem 2.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(a x^2 y+b x^3+c) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{a x^3}{3}} \left (\sqrt [3]{3} b e^{\frac{a x^3}{3}} \text{Gamma}\left (\frac{1}{3},\frac{a x^3}{3}\right )+\sqrt [3]{3} a c e^{\frac{a x^3}{3}} \text{Gamma}\left (\frac{1}{3},\frac{a x^3}{3}\right )+3 a^{4/3} y+3 \sqrt [3]{a} b x\right )}{3 a^{4/3}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,{\frac{ \left ( 3\, \WhittakerM \left ( 1/6,2/3,1/3\,a{x}^{3} \right ){{\rm e}^{1/6\,a{x}^{3}}}\sqrt [6]{3}acx+3\, \WhittakerM \left ( 1/6,2/3,1/3\,a{x}^{3} \right ){{\rm e}^{1/6\,a{x}^{3}}}\sqrt [6]{3}bx+4\,acx\sqrt [6]{a{x}^{3}}-4\,ya\sqrt [6]{a{x}^{3}} \right ){{\rm e}^{-1/3\,a{x}^{3}}}}{a\sqrt [6]{a{x}^{3}}}} \right ) \]

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37.3 problem number 3

problem number 256

Added January 2, 2019.

Problem 2.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(a x^2 y+b y^3) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{(-1)^{2/3} \left (-2^{2/3} \sqrt [3]{3} b y^2 \text{Gamma}\left (\frac{1}{3},-\frac{2 a x^3}{3}\right )+3 \sqrt [3]{a} e^{\frac{2 a x^3}{3}+\frac{i \pi }{3}}\right )}{3 \sqrt [3]{a} y^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/9\,{\frac{2\,{2}^{2/3}{3}^{5/6}bxO{y}^{2}-3\,{2}^{2/3}\sqrt [3]{3}bx\Gamma \left ( 1/3,-2/3\,a{x}^{3} \right ) \Gamma \left ( 2/3 \right ){y}^{2}+9\,O\Gamma \left ( 2/3 \right ){{\rm e}^{2/3\,a{x}^{3}}}}{O\Gamma \left ( 2/3 \right ){y}^{2}}} \right ) \]

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37.4 problem number 4

problem number 257

Added January 2, 2019.

Problem 2.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(a x y+b) y^2 w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [y^2 w^{(0,1)}(x,y) (a x y+b)+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{1}{-{b}^{2}+4\,a} \left ( 2\,\sqrt{{b}^{2}-4\,a}b\arctanh \left ({\frac{\sqrt{{b}^{2}-4\,a} \left ( 2\,axy+b \right ) }{-{b}^{2}+4\,a}} \right ) +\ln \left ({x}^{2} \left ( a{x}^{2}{y}^{2}+bxy+1 \right ) \right ){b}^{2}-2\,\ln \left ( yx \right ){b}^{2}-4\,\ln \left ({x}^{2} \left ( a{x}^{2}{y}^{2}+bxy+1 \right ) \right ) a+8\,\ln \left ( yx \right ) a \right ) } \right ) \]

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37.5 problem number 5

problem number 258

Added January 2, 2019.

Problem 2.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +A(a x+b y+c)^3 y^2 w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [A w^{(0,1)}(x,y) (a x+b y+c)^3+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,{\frac{1}{Ab}\sum _{{\it \_R}=\RootOf \left ( A{b}^{4}{{\it \_Z}}^{3}+3\,A{b}^{3}c{{\it \_Z}}^{2}+3\,A{b}^{2}{c}^{2}{\it \_Z}+Ab{c}^{3}+a \right ) }{\frac{1}{{b}^{2}{{\it \_R}}^{2}+2\,bc{\it \_R}+{c}^{2}}\ln \left ({\frac{-{\it \_R}\,b+ax+by}{b}} \right ) }}+x \right ) \] Answer contains RootOf

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37.6 problem number 6

problem number 259

Added January 2, 2019.

Problem 2.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x +(a x^4 y^3+(b x^2-1)y+c x) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a x^4 y^3+y \left (b x^2-1\right )+c x\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,b{x}^{2}+{b}^{3}\sum _{{\it \_R}=\RootOf \left ({c}^{2}a{{\it \_Z}}^{3}+{b}^{3}{\it \_Z}-{b}^{3} \right ) }{\frac{1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac{bxy+{\it \_R}\,c}{c}} \right ) } \right ) \] Answer contains RootOf

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37.7 problem number 7

problem number 260

Added January 2, 2019.

Problem 2.2.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x +(a x^2 y^2+b x y+c) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 a y x^{\sqrt{-4 a c+b^2+2 b+1}+1}+b x^{\sqrt{-4 a c+b^2+2 b+1}}+\sqrt{-4 a c+b^2+2 b+1} x^{\sqrt{-4 a c+b^2+2 b+1}}+x^{\sqrt{-4 a c+b^2+2 b+1}}}{\sqrt{-4 a c+b^2+2 b+1}-2 a x y-b-1}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{4\,ca-{b}^{2}-2\,b-1}} \left ( \ln \left ( x \right ) \sqrt{4\,ca-{b}^{2}-2\,b-1}-2\,\arctan \left ({\frac{2\,axy+b+1}{\sqrt{4\,ca-{b}^{2}-2\,b-1}}} \right ) \right ) } \right ) \]

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37.8 problem number 8

problem number 261

Added January 2, 2019.

Problem 2.2.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^2 y+b) w_x -(a x y^2+c) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,a{x}^{2}{y}^{2}-by-cx \right ) \]

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37.9 problem number 9

problem number 262

Added January 2, 2019.

Problem 2.2.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b y^3) w_x -(c x^3+a y) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,b{y}^{4}-1/4\,c{x}^{4}-axy \right ) \]