34 HFOPDE, chapter 1

 34.1 problem number 1
 34.2 problem number 2
 34.3 problem number 3
 34.4 problem number 4
 34.5 problem number 5
 34.6 problem number 6

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34.1 problem number 1

problem number 209

Added January 2, 2019.

Problem 1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x = f(x,y) \]

Mathematica

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

Maple

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

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34.2 problem number 2

problem number 210

Added January 2, 2019.

Problem 1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_y = f(x,y) \]

Mathematica

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

Maple

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

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34.3 problem number 3

problem number 211

Added January 2, 2019.

Problem 1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x = w f(x,y) \]

Mathematica

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

Maple

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

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34.4 problem number 4

problem number 212

Added January 2, 2019.

Problem 1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_y = w f(x,y) \]

Mathematica

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

Maple

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

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34.5 problem number 5

problem number 213

Added January 2, 2019.

Problem 1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x = w f(x,y)+ g(x,y) \]

Mathematica

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

Maple

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

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34.6 problem number 6

problem number 214

Added January 2, 2019.

Problem 1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_y = w f(x,y)+ g(x,y) \]

Mathematica

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

Maple

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