95 HFOPDE, chapter 4.1.1

 95.1 Example 1
 95.2 Example 2
 95.3 Example 3

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95.1 Example 1

problem number 813

Added Feb. 17, 2019.

Chapter 4.1.1 example 1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a y w_y = b y^2 w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-ax}} \right ){{\rm e}^{1/2\,{\frac{b{y}^{2}}{a}}}} \]

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95.2 Example 2

problem number 814

Added Feb. 17, 2019.

Chapter 4.1.1 example 2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a y w_y = b e^{\lambda x} y w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-ax}} \right ){{\rm e}^{{\frac{by{{\rm e}^{-ax+x \left ( a+\lambda \right ) }}}{a+\lambda }}}} \]

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95.3 Example 3

problem number 815

Added Feb. 17, 2019.

Chapter 4.1.1 example 3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = b w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ){{\rm e}^{bx}} \]