93 HFOPDE, chapter 3.8.3

 93.1 Problem 1
 93.2 Problem 2
 93.3 Problem 3
 93.4 Problem 4
 93.5 Problem 5
 93.6 Problem 6
 93.7 Problem 7
 93.8 Problem 8

_______________________________________________________________________________________

93.1 Problem 1

problem number 798

Added Feb. 11, 2019.

Problem Chapter 3.8.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = f(\alpha x+\beta y) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \int _1^x \frac{f\left (\frac{\beta (b K[1]+a y-b x)}{a}+\alpha K[1]\right )}{a} \, dK[1]+c_1\left (\frac{a y-b x}{a}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{a}f \left ({\frac{ \left ( ay-bx \right ) \beta +{\it \_a}\,a\alpha +b\beta \,{\it \_a}}{a}} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

_______________________________________________________________________________________

93.2 Problem 2

problem number 799

Added Feb. 11, 2019.

Problem Chapter 3.8.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = x f(\frac{y}{x}) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{y}{x}\right )+x f\left (\frac{y}{x}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =xf \left ({\frac{y}{x}} \right ) +{\it \_F1} \left ({\frac{y}{x}} \right ) \]

_______________________________________________________________________________________

93.3 Problem 3

problem number 800

Added Feb. 11, 2019.

Problem Chapter 3.8.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = f(x^2+y^2) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{{\it \_a}}f \left ({\frac{{y}^{2}{{\it \_a}}^{2}}{{x}^{2}}}+{{\it \_a}}^{2} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{y}{x}} \right ) \]

_______________________________________________________________________________________

93.4 Problem 4

problem number 801

Added Feb. 11, 2019.

Problem Chapter 3.8.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = x f(\frac{y}{x})+ g(x^2+y^2) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{{\it \_a}} \left ({\it \_a}\,f \left ({\frac{y}{x}} \right ) +g \left ({\frac{{y}^{2}{{\it \_a}}^{2}}{{x}^{2}}}+{{\it \_a}}^{2} \right ) \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{y}{x}} \right ) \]

_______________________________________________________________________________________

93.5 Problem 5

problem number 802

Added Feb. 11, 2019.

Problem Chapter 3.8.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = x^k f(x^n y^m) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{{{\it \_a}}^{k-1}}{a}f \left ({{\it \_a}}^{n} \left ( y{x}^{-{\frac{b}{a}}}{{\it \_a}}^{{\frac{b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ) \]

_______________________________________________________________________________________

93.6 Problem 6

problem number 803

Added Feb. 11, 2019.

Problem Chapter 3.8.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ m x w_x + n y w_y = f(a x^n + b y^m) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{m{\it \_a}}f \left ( a{{\it \_a}}^{n}+b \left ( y{x}^{-{\frac{n}{m}}}{{\it \_a}}^{{\frac{n}{m}}} \right ) ^{m} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac{n}{m}}} \right ) \]

_______________________________________________________________________________________

93.7 Problem 7

problem number 804

Added Feb. 17, 2019.

Problem Chapter 3.8.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + x y w_y = y^k f(\alpha x + \beta y) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{{{\it \_a}}^{2}}f \left ({\it \_a}\, \left ({\frac{\beta \,y}{x}}+\alpha \right ) \right ) \left ({\frac{y{\it \_a}}{x}} \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ({\frac{y}{x}} \right ) \]

_______________________________________________________________________________________

93.8 Problem 8

problem number 805

Added Feb. 17, 2019.

Problem Chapter 3.8.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \frac{f(x)}{f'(x)} w_x + \frac{g(y)}{g'(y)} w_y = h(f(x)+g(y)) \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\log \left (\frac{\text{InverseFunction}\left [g^{(-1)},1,1\right ][y]}{f(x)}\right )\right )+\int _1^x \frac{f'(K[1]) h\left (g\left (\text{InverseFunction}\left [\text{InverseFunction}\left [g^{(-1)},1,1\right ],1,1\right ]\left [\frac{f(K[1]) \text{InverseFunction}\left [g^{(-1)},1,1\right ][y]}{f(x)}\right ]\right )+f(K[1])\right )}{f(K[1])} \, dK[1]\right \}\right \} \]

Maple

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