65 HFOPDE, chapter 2.9.2

 65.1 problem number 1
 65.2 problem number 2
 65.3 problem number 3
 65.4 problem number 4
 65.5 problem number 5
 65.6 problem number 6
 65.7 problem number 7
 65.8 problem number 8
 65.9 problem number 9
 65.10 problem number 10
 65.11 problem number 11
 65.12 problem number 12
 65.13 problem number 13
 65.14 problem number 14
 65.15 problem number 15
 65.16 problem number 16

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65.1 problem number 1

problem number 597

Added Feb. 7, 2019.

Problem 2.9.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + f(a x+b y + c) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) f(a x+b y+c)+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.2 problem number 2

problem number 598

Added Feb. 7, 2019.

Problem 2.9.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + f(\frac{y}{x}) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [f\left (\frac{y}{x}\right ) w^{(0,1)}(x,y)+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.3 problem number 3

problem number 599

Added Feb. 7, 2019.

Problem 2.9.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(y+a x^n+b) - a n x^{n-1} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (f\left (a x^n+b+y\right )-a n x^{n-1}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.4 problem number 4

problem number 600

Added Feb. 7, 2019.

Problem 2.9.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y f(x^n y^m) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [y w^{(0,1)}(x,y) f\left (y^m x^n\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.5 problem number 5

problem number 601

Added Feb. 7, 2019.

Problem 2.9.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ y^{m-1} w_x + x^{n-1} f(a x^n+b y^m) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [x^{n-1} w^{(0,1)}(x,y) f\left (a x^n+b y^m\right )+y^{m-1} w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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65.6 problem number 6

problem number 602

Added Feb. 7, 2019.

Problem 2.9.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{-\lambda x} f(e^{\lambda x} y) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [e^{-\lambda x} w^{(0,1)}(x,y) f\left (y e^{\lambda x}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.7 problem number 7

problem number 603

Added Feb. 7, 2019.

Problem 2.9.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{\lambda y} f(e^{\lambda y} x) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [e^{\lambda y} w^{(0,1)}(x,y) f\left (x e^{\lambda y}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.8 problem number 8

problem number 604

Added Feb. 7, 2019.

Problem 2.9.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + y f(e^{\alpha x} y^m) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [y w^{(0,1)}(x,y) f\left (e^{\alpha x} y^m\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.9 problem number 9

problem number 605

Added Feb. 7, 2019.

Problem 2.9.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + f(x^n e^{\alpha y}) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) f\left (e^{\alpha y} x^n\right )+x w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.10 problem number 10

problem number 606

Added Feb. 7, 2019.

Problem 2.9.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{\lambda x - \beta y} f(a e^{\lambda x} + b e^{\beta y}) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) e^{\lambda x-\beta y} f\left (a e^{\lambda x}+b e^{\beta y}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\lambda } \left ( \int ^{-{\frac{a{{\rm e}^{\lambda \,x}}+b{{\rm e}^{\beta \,y}}}{\lambda \,a}}}\! \left ( bf \left ( -\lambda \,a{\it \_a} \right ) \beta +\lambda \,a \right ) ^{-1}{d{\it \_a}}a{\lambda }^{2}+{{\rm e}^{\lambda \,x}} \right ) } \right ) \]

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65.11 problem number 11

problem number 607

Added Feb. 7, 2019.

Problem 2.9.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f\left (y+a e^{\lambda x}+b \right ) -a \lambda e^{\lambda x} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (f\left (a e^{\lambda x}+b+y\right )-a \lambda e^{\lambda x}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.12 problem number 12

problem number 608

Added Feb. 7, 2019.

Problem 2.9.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \alpha x y w_x + \left ( \alpha f(x^n e^{\alpha y}) - n y \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (\alpha f\left (e^{\alpha y} x^n\right )-n y\right )+\alpha x y w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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65.13 problem number 13

problem number 609

Added Feb. 7, 2019.

Problem 2.9.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ m x(\ln y) w_x + \left ( y f(x^n y^m) - n y \ln y \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (y f\left (y^m x^n\right )-n y \log (y)\right )+m x \log (y) w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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65.14 problem number 14

problem number 610

Added Feb. 7, 2019.

Problem 2.9.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(y+a \tan x) - a \tan ^2 x \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (f(a \tan (x)+y)-a \tan ^2(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.15 problem number 15

problem number 611

Added Feb. 7, 2019.

Problem 2.9.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ e^{\lambda x} w_x + f(\lambda x+\ln y) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) f(\lambda x+\log (y))+e^{\lambda x} w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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65.16 problem number 16

problem number 612

Added Feb. 7, 2019.

Problem 2.9.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + e^{\lambda y} f(\lambda y+\ln x) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [e^{\lambda y} w^{(0,1)}(x,y) f(\lambda y+\log (x))+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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