66 HFOPDE, chapter 2.9.3

 66.1 problem number 1
 66.2 problem number 2
 66.3 problem number 3
 66.4 problem number 4
 66.5 problem number 5
 66.6 problem number 6
 66.7 problem number 7
 66.8 problem number 8
 66.9 problem number 9
 66.10 problem number 11
 66.11 problem number 12
 66.12 problem number 13
 66.13 problem number 14
 66.14 problem number 15
 66.15 problem number 16
 66.16 problem number 17
 66.17 problem number 18
 66.18 problem number 19
 66.19 problem number 20
 66.20 problem number 21
 66.21 problem number 22
 66.22 problem number 23

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66.1 problem number 1

problem number 613

Added Feb. 7, 2019.

Problem 2.9.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int _{{\it \_b}}^{x}\!{\frac{1}{g \left ({{\it \_a}}^{n}{y}^{m} \right ) } \left ( n{{\it \_a}}^{-{\frac{kn+m-n}{m}}}{y}^{-k+1}-{{\it \_a}}^{-{\frac{n \left ( k-1 \right ) }{m}}}f \left ({\it \_a} \right ) g \left ({{\it \_a}}^{n}{y}^{m} \right ) \right ) }\,{\rm d}{\it \_a}-\int \!{\frac{{y}^{-k}}{g \left ({x}^{n}{y}^{m} \right ) } \left ( g \left ({x}^{n}{y}^{m} \right ) \int _{{\it \_b}}^{x}\!{\frac{1}{ \left ( g \left ({{\it \_a}}^{n}{y}^{m} \right ) \right ) ^{2}} \left ({{\it \_a}}^{-{\frac{kn-mn+m-n}{m}}}\mbox{D} \left ( g \right ) \left ({{\it \_a}}^{n}{y}^{m} \right ) m{y}^{m}+{{\it \_a}}^{-{\frac{kn+m-n}{m}}}g \left ({{\it \_a}}^{n}{y}^{m} \right ) k-{{\it \_a}}^{-{\frac{kn+m-n}{m}}}g \left ({{\it \_a}}^{n}{y}^{m} \right ) \right ) }\,{\rm d}{\it \_a}n+m{x}^{-{\frac{n \left ( k-1 \right ) }{m}}} \right ) }\,{\rm d}y \right ) \]

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66.2 problem number 2

problem number 614

Added Feb. 9, 2019.

Problem 2.9.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.3 problem number 3

problem number 615

Added Feb. 9, 2019.

Problem 2.9.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( f(\frac{y}{x})+x^\alpha h(\frac{y}{x}) \right ) w_x + \left ( g(\frac{y}{x})+ y x^{\alpha -1} h(\frac{y}{x}) \right ) w_y = 0 \]

Mathematica

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

Maple

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

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66.4 problem number 4

problem number 616

Added Feb. 9, 2019.

Problem 2.9.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( f(a x+b y)+ b x g(a x+b y) \right ) w_x + \left ( h(a x+b y) - a x g(a x+b y) \right ) w_y = 0 \]

Mathematica

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

Maple

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

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66.5 problem number 5

problem number 617

Added Feb. 9, 2019.

Problem 2.9.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( f(a x+b y)+ b y g(a x+b y) \right ) w_x + \left ( h(a x+b y) - a y g(a x+b y) \right ) w_y = 0 \]

Mathematica

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

Maple

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

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66.6 problem number 6

problem number 618

Added Feb. 9, 2019.

Problem 2.9.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.7 problem number 7

problem number 619

Added Feb. 9, 2019.

Problem 2.9.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.8 problem number 8

problem number 620

Added Feb. 9, 2019.

Problem 2.9.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.9 problem number 9

problem number 621

Added Feb. 9, 2019.

Problem 2.9.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke 1965.

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

\[ f_y *w_x - f_x w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1(\text{InverseFunction}[\text{InverseFunction}[f,2,2],2,2][x,y])\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -f \left ( x,y \right ) \right ) \]

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66.10 problem number 11

problem number 622

Added Feb. 9, 2019.

Problem 2.9.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke 1965.

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

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

Mathematica

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

Maple

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

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66.11 problem number 12

problem number 623

Added Feb. 9, 2019.

Problem 2.9.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke 1965.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{m} \left ( -\int \!{\frac{{y}^{-k}}{mg \left ({{\rm e}^{\alpha \,x}}{y}^{m} \right ) } \left ( \alpha \,\int _{{\it \_b}}^{x}\!{\frac{1}{ \left ( g \left ({{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) \right ) ^{2}} \left ({{\rm e}^{-{\frac{{\it \_a}\,\alpha \, \left ( k-m-1 \right ) }{m}}}}m{y}^{m}\mbox{D} \left ( g \right ) \left ({{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) +{{\rm e}^{-{\frac{\alpha \,{\it \_a}\, \left ( k-1 \right ) }{m}}}}kg \left ({{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) -{{\rm e}^{-{\frac{\alpha \,{\it \_a}\, \left ( k-1 \right ) }{m}}}}g \left ({{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) \right ) }\,{\rm d}{\it \_a}g \left ({{\rm e}^{\alpha \,x}}{y}^{m} \right ) +{{\rm e}^{-{\frac{\alpha \,x \left ( k-1 \right ) }{m}}}}m \right ) }\,{\rm d}ym+\int _{{\it \_b}}^{x}\!{\frac{mf \left ({\it \_a} \right ) g \left ({{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) -{y}^{-k+1}\alpha }{g \left ({{\rm e}^{{\it \_a}\,\alpha }}{y}^{m} \right ) }{{\rm e}^{-{\frac{\alpha \,{\it \_a}\, \left ( k-1 \right ) }{m}}}}}\,{\rm d}{\it \_a} \right ) } \right ) \]

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66.12 problem number 13

problem number 624

Added Feb. 9, 2019.

Problem 2.9.3.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.13 problem number 14

problem number 625

Added Feb. 9, 2019.

Problem 2.9.3.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.14 problem number 15

problem number 626

Added Feb. 9, 2019.

Problem 2.9.3.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.15 problem number 16

problem number 627

Added Feb. 9, 2019.

Problem 2.9.3.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.16 problem number 17

problem number 628

Added Feb. 9, 2019.

Problem 2.9.3.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.17 problem number 18

problem number 629

Added Feb. 9, 2019.

Problem 2.9.3.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.18 problem number 19

problem number 630

Added Feb. 9, 2019.

Problem 2.9.3.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.19 problem number 20

problem number 631

Added Feb. 9, 2019.

Problem 2.9.3.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cos y w_x + \left ( f(x) g(\sin x \sin y) - \cot x \sin y \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) (f(x) g(\sin (x) \sin (y))-\cot (x) \sin (y))+\cos (y) w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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66.20 problem number 21

problem number 632

Added Feb. 9, 2019.

Problem 2.9.3.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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66.21 problem number 22

problem number 633

Added Feb. 9, 2019.

Problem 2.9.3.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( x \cos ^2 y f(x) g(x^{2 n} \tan y) - n \sin 2 y \right ) w_y = 0 \]

Mathematica

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

Maple

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

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66.22 problem number 23

problem number 634

Added Feb. 9, 2019.

Problem 2.9.3.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \cos ^2 y f(x) g(e^{2 x} \tan y) -\sin 2 y \right ) w_y = 0 \]

Mathematica

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

Maple

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