107 HFOPDE, chapter 4.5.1

 107.1 Problem 1
 107.2 Problem 2
 107.3 Problem 3
 107.4 Problem 4
 107.5 Problem 5
 107.6 Problem 6

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107.1 Problem 1

problem number 895

Added Feb. 25, 2019.

Problem Chapter 4.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \ln (\lambda x + \beta y) w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \left ( \beta \,y+\lambda \,x \right ) ^{{\frac{c \left ( \beta \,y+\lambda \,x \right ) }{\lambda \,a+b\beta }}}{{\rm e}^{-{\frac{c \left ( \beta \,y+\lambda \,x \right ) }{\lambda \,a+b\beta }}}} \]

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107.2 Problem 2

problem number 896

Added Feb. 25, 2019.

Problem Chapter 4.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c \ln (\lambda x)+ k \ln (\beta y) \right ) w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \left ( \lambda \,x \right ) ^{{\frac{cx}{a}}} \left ( \beta \,y \right ) ^{{\frac{ky}{b}}}{{\rm e}^{{\frac{-aky-bcx}{ab}}}} \]

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107.3 Problem 3

problem number 897

Added Feb. 25, 2019.

Problem Chapter 4.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \ln ^n(\lambda x) w_y = \left ( c \ln ^m(\mu x)+ s \ln ^k(\beta y) \right ) w \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac{b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( c \left ( \ln \left ( \lambda \,{\it \_b} \right ) \right ) ^{m}+s \left ( \ln \left ( \beta \, \left ( \int \!{\frac{b \left ( \ln \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}-\int \!{\frac{b \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]

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107.4 Problem 4

problem number 898

Added Feb. 25, 2019.

Problem Chapter 4.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \ln ^n(\lambda y) w_y = \left ( c \ln ^m(\mu x)+ s \ln ^k(\beta y) \right ) w \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{a\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ){{\rm e}^{\int ^{y}\!{\frac{ \left ( \ln \left ( \lambda \,{\it \_b} \right ) \right ) ^{-n}}{b} \left ( c \left ( \ln \left ( \lambda \, \left ( \int \!{\frac{ \left ( \ln \left ( \lambda \,{\it \_b} \right ) \right ) ^{-n}a}{b}}\,{\rm d}{\it \_b}-{\frac{a\int \! \left ( \ln \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \right ) ^{m}+s \left ( \ln \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]

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107.5 Problem 5

problem number 899

Added Feb. 25, 2019.

Problem Chapter 4.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \ln (\beta y) w_x + a \ln (\lambda x) w_y = b w \ln (\beta y) \]

Mathematica

\[ \left \{\left \{w(x,y)\to e^{b x} c_1\left (y \left (\log \left (\beta y e^{\frac{a x}{y}} x^{-\frac{a x}{y}} \lambda ^{-\frac{a x}{y}}\right )-1\right )\right )\right \}\right \} \]

Maple

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

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107.6 Problem 6

problem number 900

Added Feb. 25, 2019.

Problem Chapter 4.5.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \ln (\lambda x)^n w_x + b \ln (\beta y)^k w_y = c \ln (\gamma x)^m w \]

Mathematica

\[ \text{DSolve}\left [a w^{(1,0)}(x,y) \log ^n(\lambda x)+b w^{(0,1)}(x,y) \log ^k(\beta y)=c w(x,y) \log ^m(\gamma x),w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}\,{\rm d}x+\int \!{\frac{ \left ( \ln \left ( \beta \,y \right ) \right ) ^{-k}a}{b}}\,{\rm d}y \right ){{\rm e}^{\int \!{\frac{c \left ( \ln \left ( \gamma \right ) +\ln \left ( x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}x}} \]