79 HFOPDE, chapter 3.5.1

 79.1 Problem 1
 79.2 Problem 2
 79.3 Problem 3
 79.4 Problem 4
 79.5 Problem 5
 79.6 Problem 6

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79.1 Problem 1

problem number 717

Added Feb. 11, 2019.

Problem Chapter 3.5.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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79.2 Problem 2

problem number 718

Added Feb. 11, 2019.

Problem Chapter 3.5.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{ \left ( \ln \left ( \lambda \,x \right ) bc-bc \right ) x}{ab}}+{\frac{y}{ab} \left ( \ln \left ({\frac{b\beta \,x}{a}}+{\frac{ \left ( ay-bx \right ) \beta }{a}} \right ) ak-ka \right ) }+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

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79.3 Problem 3

problem number 719

Added Feb. 11, 2019.

Problem Chapter 3.5.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{c \left (a \lambda \text{LogIntegral}(\beta y)-b \beta \left (\log \left (e^{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )-1\right ) \text{ExpIntegralEi}\left (\log \left (e^{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )\right ) \left (\log \left (\frac{\gamma x (\log (\lambda x)-1)}{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac{\lambda x (\log (\lambda x)-1)}{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )+b \beta e^{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )+1} \left (\log \left (\frac{\gamma x (\log (\lambda x)-1)}{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac{\lambda x (\log (\lambda x)-1)}{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )\right )+a b \beta \lambda c_1\left (\frac{\text{LogIntegral}(\beta y)}{\beta }-\frac{b x (\log (\lambda x)-1)}{a}\right )+b \beta c \text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right ) \text{ExpIntegralEi}\left (\log \left (e^{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )+1}\right )\right ) \left (\log \left (\frac{\gamma x (\log (\lambda x)-1)}{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )}\right )-\log \left (\frac{\lambda x (\log (\lambda x)-1)}{\text{ProductLog}\left (\frac{\lambda x (\log (\lambda x)-1)}{e}\right )}\right )\right )}{a b \beta \lambda }\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\lambda \,a\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) } \left ( \LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ){\it \_F1} \left ({\frac{a\Ei \left ( 1,-\ln \left ( \beta \,y \right ) \right ) +b\beta \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{a\beta }} \right ) a\lambda + \left ( -\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) \left ( 1-\ln \left ({\frac{\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) }} \right ) +\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) \right ) \left ( \ln \left ( \gamma \right ) +\ln \left ({\frac{x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) }} \right ) -\ln \left ({\frac{\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) }} \right ) \right ) \Ei \left ( 1,-\ln \left ({\frac{\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) }} \right ) \right ) +\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) \left ( \ln \left ( \gamma \right ) +\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) +\ln \left ({\frac{x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) }} \right ) -\ln \left ({\frac{\lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ) }{\LambertW \left ( \lambda \,x \left ( \ln \left ( \lambda \,x \right ) -1 \right ){{\rm e}^{-1}} \right ) }} \right ) \right ) \right ) c \right ) } \]

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79.4 Problem 4

problem number 720

Added Feb. 11, 2019.

Problem Chapter 3.5.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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79.5 Problem 5

problem number 721

Added Feb. 11, 2019.

Problem Chapter 3.5.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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79.6 Problem 6

problem number 722

Added Feb. 11, 2019.

Problem Chapter 3.5.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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