67 HFOPDE, chapter 3 examples

 67.1 Example 1
 67.2 Example 2
 67.3 Example 3

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67.1 Example 1

problem number 635

Added Feb. 9, 2019.

Problem Chapter 3, example 1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c \]

Mathematica

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

Maple

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

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67.2 Example 2

problem number 636

Added Feb. 9, 2019.

Problem Chapter 3, example 2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^x w_x + b w_y = c e^{2 x} y \]

Mathematica

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

Maple

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

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67.3 Example 3

problem number 637

Added Feb. 9, 2019.

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

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

\[ w_x + a w_y = b \]

Mathematica

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

Maple

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