6.4.26 8.3

   6.4.26.1 [1183] Problem 1
   6.4.26.2 [1184] Problem 2
   6.4.26.3 [1185] Problem 3
   6.4.26.4 [1186] Problem 4
   6.4.26.5 [1187] Problem 5
   6.4.26.6 [1188] Problem 6
   6.4.26.7 [1189] Problem 7

6.4.26.1 [1183] Problem 1

problem number 1183

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

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

Maple

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

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6.4.26.2 [1184] Problem 2

problem number 1184

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

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

Maple

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

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6.4.26.3 [1185] Problem 3

problem number 1185

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

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

Maple

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

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6.4.26.4 [1186] Problem 4

problem number 1186

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

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

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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6.4.26.5 [1187] Problem 5

problem number 1187

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

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

Maple

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

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6.4.26.6 [1188] Problem 6

problem number 1188

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

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

Maple

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

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6.4.26.7 [1189] Problem 7

problem number 1189

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

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

Maple

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

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