125 HFOPDE, chapter 5.2.4

 125.1 Problem 1
 125.2 Problem 2

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125.1 Problem 1

problem number 1012

Added March 12, 2019.

Problem Chapter 5.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + k x^n y^m \]

Mathematica

\[ \left \{\left \{w(x,y)\to e^{\frac{c x}{a}} \left (\int _1^x \frac{k K[1]^n e^{-\frac{c K[1]}{a}} \left (\frac{b K[1]+a y-b x}{a}\right )^m}{a} \, dK[1]+c_1\left (\frac{a y-b x}{a}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{k{{\it \_a}}^{n}}{a} \left ({\frac{{\it \_a}\,b+ay-bx}{a}} \right ) ^{m}{{\rm e}^{-{\frac{c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \right ){{\rm e}^{{\frac{cx}{a}}}} \]

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125.2 Problem 2

problem number 1013

Added March 12, 2019.

Problem Chapter 5.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{c{{\it \_a}}^{n}}{a} \left ( y{{\rm e}^{-{\frac{x}{a}}+{\frac{{\it \_a}}{a}}}} \right ) ^{m}{{\rm e}^{-{\frac{{\it \_a}\,b}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac{x}{a}}}} \right ) \right ){{\rm e}^{{\frac{bx}{a}}}} \]

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