120 HFOPDE, chapter 4.8.3

 120.1 Problem 1
 120.2 Problem 2
 120.3 Problem 3
 120.4 Problem 4
 120.5 Problem 5
 120.6 Problem 6
 120.7 Problem 7

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120.1 Problem 1

problem number 975

Added March 10, 2019.

Problem Chapter 4.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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120.2 Problem 2

problem number 976

Added March 10, 2019.

Problem Chapter 4.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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120.3 Problem 3

problem number 977

Added March 10, 2019.

Problem Chapter 4.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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120.4 Problem 4

problem number 978

Added March 10, 2019.

Problem Chapter 4.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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120.5 Problem 5

problem number 979

Added March 10, 2019.

Problem Chapter 4.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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120.6 Problem 6

problem number 980

Added March 10, 2019.

Problem Chapter 4.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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120.7 Problem 7

problem number 981

Added March 10, 2019.

Problem Chapter 4.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y-\int _1^x \frac{g(K[1]) f'(K[1])}{f(K[1]) g'(K[1])} \, dK[1]\right ) \exp \left (\int _1^x \frac{f'(K[2]) h\left (g\left (-\text{Integrate}\left [\frac{g(K[1]) f'(K[1])}{f(K[1]) g'(K[1])},\{K[1],1,x\},\text{Assumptions}\to \text{True}\right ]+\text{Integrate}\left [\frac{g(K[1]) f'(K[1])}{f(K[1]) g'(K[1])},\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}\right ]+y\right )+f(K[2])\right )}{f(K[2])} \, dK[2]\right )\right \}\right \} \]

Maple

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