122 HFOPDE, chapter 5.2.1

 122.1 Problem 1
 122.2 Problem 2
 122.3 Problem 3
 122.4 Problem 4
 122.5 Problem 5
 122.6 Problem 6
 122.7 Problem 7
 122.8 Problem 8

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122.1 Problem 1

problem number 989

Added March 10, 2019.

Problem Chapter 5.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + d \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{c} \left ({{\rm e}^{{\frac{cx}{a}}}}{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) c-d \right ) } \]

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122.2 Problem 2

problem number 990

Added March 10, 2019.

Problem Chapter 5.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (x-a) w_x + (y-b) w_y = w - c \]

Mathematica

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

Maple

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

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122.3 Problem 3

problem number 991

Added March 10, 2019.

Problem Chapter 5.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b) w_x + (c x+d) w_y = \alpha w + \beta \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\alpha } \left ( \left ( ax+b \right ) ^{{\frac{\alpha }{a}}}{\it \_F1} \left ({\frac{y{a}^{2}-\ln \left ( ax+b \right ) da+\ln \left ( ax+b \right ) bc-cxa}{{a}^{2}}} \right ) \alpha -\beta \right ) } \]

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122.4 Problem 4

problem number 992

Added March 10, 2019.

Problem Chapter 5.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\alpha } \left ( \left ( ax+b \right ) ^{{\frac{\alpha }{a}}}{\it \_F1} \left ({\frac{cy+d}{c} \left ( ax+b \right ) ^{-{\frac{c}{a}}}} \right ) \alpha -\beta \right ) } \]

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122.5 Problem 5

problem number 993

Added March 10, 2019.

Problem Chapter 5.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{ \left ( -\alpha +c \right ) \left ( a-\alpha \right ) \alpha } \left ( \left ( ax+b \right ) ^{{\frac{\alpha }{a}}}\alpha \, \left ( -\alpha +c \right ) \left ( a-\alpha \right ){\it \_F1} \left ({\frac{cy+d}{c} \left ( ax+b \right ) ^{-{\frac{c}{a}}}} \right ) + \left ( -\beta \,y-\gamma \,x \right ){\alpha }^{2}+ \left ( a\beta \,y+\gamma \,xc-b\gamma -\beta \,d \right ) \alpha +a\beta \,d+bc\gamma \right ) } \]

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122.6 Problem 6

problem number 994

Added March 10, 2019.

Problem Chapter 5.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\alpha } \left ( \left ( ax+b \right ) ^{{\frac{\alpha }{a}}}{\it \_F1} \left ({\frac{dya-cxd-{d}^{2}y-bc}{ \left ( a-d \right ) d} \left ( ax+b \right ) ^{-{\frac{d}{a}}}} \right ) \alpha -\beta \right ) } \]

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122.7 Problem 7

problem number 995

Added March 10, 2019.

Problem Chapter 5.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a_1 x+a_0) w_x + (b_2 y+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_2 y+k_1 x+k_0 \]

Mathematica

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

Maple

\[ \text{ sol=() } \] Timed out

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122.8 Problem 8

problem number 996

Added March 10, 2019.

Problem Chapter 5.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + (b_1 x+b_0) w_y = (c_1 x+c_0) w + s_1 x+s0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{{\it \_a}\,{\it s1}+{\it s0}}{\sqrt{a \left ({{\it \_a}}^{2}{\it b1}+{y}^{2}a-{\it b1}\,{x}^{2}+2\,{\it \_a}\,{\it b0}-2\,{\it b0}\,x \right ) }} \left ({\frac{a{\it b1}\,{\it \_a}+\sqrt{a \left ({{\it \_a}}^{2}{\it b1}+{y}^{2}a-{\it b1}\,{x}^{2}+2\,{\it \_a}\,{\it b0}-2\,{\it b0}\,x \right ) }\sqrt{a{\it b1}}+a{\it b0}}{\sqrt{a{\it b1}}}} \right ) ^{{\frac{{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{{\it b1}\,\sqrt{a{\it b1}}}}}{{\rm e}^{-{\frac{{\it c1}\,\sqrt{a \left ({{\it \_a}}^{2}{\it b1}+{y}^{2}a-{\it b1}\,{x}^{2}+2\,{\it \_a}\,{\it b0}-2\,{\it b0}\,x \right ) }}{a{\it b1}}}}}}{d{\it \_a}}+{\it \_F1} \left ({\frac{{y}^{2}a-{\it b1}\,{x}^{2}-2\,{\it b0}\,x}{a}} \right ) \right ) \left ({\frac{a{\it b1}\,x+a{\it b0}}{\sqrt{a{\it b1}}}}+\sqrt{a{\it b1}\,{x}^{2}+ \left ({y}^{2}a-{\it b1}\,{x}^{2}-2\,{\it b0}\,x \right ) a+2\,a{\it b0}\,x} \right ) ^{-{\frac{{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{{\it b1}\,\sqrt{a{\it b1}}}}}{{\rm e}^{{\frac{{\it c1}\,\sqrt{a{\it b1}\,{x}^{2}+ \left ({y}^{2}a-{\it b1}\,{x}^{2}-2\,{\it b0}\,x \right ) a+2\,a{\it b0}\,x}}{a{\it b1}}}}} \]