62 HFOPDE, chapter 2.8.5

 62.1 problem number 1
 62.2 problem number 2
 62.3 problem number 3
 62.4 problem number 4
 62.5 problem number 5

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62.1 problem number 1

problem number 575

Added Feb. 7, 2019.

Problem 2.8.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({(\cos \left ( \lambda \,x \right ) y-1) \left ( -\cos \left ( \lambda \,x \right ) \int \!-\lambda \,{{\rm e}^{\int \!{\frac{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\cos \left ( \lambda \,x \right ) \sin \left ( \lambda \,x \right ) }}\,{\rm d}x}}\sin \left ( \lambda \,x \right ) \,{\rm d}xy+{{\rm e}^{\int \!{\frac{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\cos \left ( \lambda \,x \right ) \sin \left ( \lambda \,x \right ) }}\,{\rm d}x}}\cos \left ( \lambda \,x \right ) +\int \!-\lambda \,{{\rm e}^{\int \!{\frac{ \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt{ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\cos \left ( \lambda \,x \right ) \sin \left ( \lambda \,x \right ) }}\,{\rm d}x}}\sin \left ( \lambda \,x \right ) \,{\rm d}x \right ) ^{-1}} \right ) \]

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62.2 problem number 2

problem number 576

Added Feb. 7, 2019.

Problem 2.8.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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62.3 problem number 3

problem number 577

Added Feb. 7, 2019.

Problem 2.8.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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62.4 problem number 4

problem number 578

Added Feb. 7, 2019.

Problem 2.8.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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62.5 problem number 5

problem number 579

Added Feb. 7, 2019.

Problem 2.8.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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