64 HFOPDE, chapter 2.9.1

 64.1 problem number 1
 64.2 problem number 2
 64.3 problem number 3
 64.4 problem number 4
 64.5 problem number 5

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64.1 problem number 1

problem number 592

Added Feb. 7, 2019.

Problem 2.9.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + g(y) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \frac{1}{g(K[1])} \, dK[1]-\int _1^x \frac{1}{f(K[2])} \, dK[2]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \]

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64.2 problem number 2

problem number 593

Added Feb. 7, 2019.

Problem 2.9.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (f(x)+g(y)) w_x + f'(x) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [f'(x) w^{(0,1)}(x,y)+w^{(1,0)}(x,y) (f(x)+g(y))=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{{\rm e}^{-y}}g \left ( y \right ) \,{\rm d}y+{{\rm e}^{-y}}f \left ( x \right ) \right ) \]

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64.3 problem number 3

problem number 594

Added Feb. 7, 2019.

Problem 2.9.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (x^n f(y) + x g(y)) w_x + h(y) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (f(y) x^n+x g(y)\right )+h(y) w^{(0,1)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({x}^{1-n}{{\rm e}^{ \left ( n-1 \right ) \int \!{\frac{g \left ( y \right ) }{h \left ( y \right ) }}\,{\rm d}y}}+n\int \!{\frac{f \left ( y \right ) }{h \left ( y \right ) }{{\rm e}^{ \left ( n-1 \right ) \int \!{\frac{g \left ( y \right ) }{h \left ( y \right ) }}\,{\rm d}y}}}\,{\rm d}y-\int \!{\frac{f \left ( y \right ) }{h \left ( y \right ) }{{\rm e}^{ \left ( n-1 \right ) \int \!{\frac{g \left ( y \right ) }{h \left ( y \right ) }}\,{\rm d}y}}}\,{\rm d}y \right ) \]

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64.4 problem number 4

problem number 595

Added Feb. 7, 2019.

Problem 2.9.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (f(y) + a m x^n y^{m-1}) w_x - (g(x)+a n x^{n-1} y^m) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a m y^{m-1} x^n+f(y)\right )-w^{(0,1)}(x,y) \left (a n y^m x^{n-1}+g(x)\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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64.5 problem number 5

problem number 596

Added Feb. 7, 2019.

Problem 2.9.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (e^{\alpha x} f(y) + c \beta ) w_x - (e^{\beta y} g(x) + c \alpha ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (e^{\alpha x} f(y)+\beta c\right )-w^{(0,1)}(x,y) \left (\alpha c+e^{\beta y} g(x)\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

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