53 HFOPDE, chapter 2.6.5

 53.1 problem number 1
 53.2 problem number 2
 53.3 problem number 3
 53.4 problem number 4
 53.5 problem number 5
 53.6 problem number 6
 53.7 problem number 7
 53.8 problem number 8
 53.9 problem number 9
 53.10 problem number 10
 53.11 problem number 11

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53.1 problem number 1

problem number 485

Added January 20, 2019.

Problem 2.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \sin ^k(\lambda x) \cos ^n(\mu y) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{a \cos (\lambda x) \sin ^{k+1}(\lambda x) \sin ^2(\lambda x)^{-\frac{k}{2}-\frac{1}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-k}{2},\frac{3}{2},\cos ^2(\lambda x)\right )+\frac{\lambda \sqrt{\sin ^2(\mu y)} \csc (\mu y) \cos ^{1-n}(\mu y) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-n}{2},\frac{3-n}{2},\cos ^2(\mu y)\right )}{\mu (n-1)}}{\lambda }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac{ \left ( \cos \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] Has unresolved integrals

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53.2 problem number 2

problem number 486

Added January 20, 2019.

Problem 2.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2-y \tan x+a(1-a) \cot ^2 x \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 y \cos ^2(x) \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}-2 y \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}-i \sqrt{-\frac{1}{(a-1) a}-4} \sqrt{a-1} \sqrt{a} \sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}-\sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}}{-2 y \cos ^2(x) \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (-\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}+2 y \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (-\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}-i \sqrt{-\frac{1}{(a-1) a}-4} \sqrt{a-1} \sqrt{a} \sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (-\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}+\sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac{1}{4} i \left (-\sqrt{-\frac{1}{(a-1) a}-4}-\frac{i}{\sqrt{a-1} \sqrt{a}}\right ) \sqrt{a-1} \sqrt{a}}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{-1+2\,a} \left ( y\sin \left ( x \right ) +\cos \left ( x \right ) a \right ) }{y\sin \left ( x \right ) -\cos \left ( x \right ) a+\cos \left ( x \right ) }} \right ) \]

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53.3 problem number 3

problem number 487

Added January 20, 2019.

Problem 2.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2-m y \tan x+b^2 \cos ^{2 m} x \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{\sqrt{b^2} \sqrt{\sin ^2(x)} \csc (x) \cos ^{m+1}(x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},\cos ^2(x)\right )}{m+1}+\tan ^{-1}\left (\frac{y \cos ^{-m}(x)}{\sqrt{b^2}}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\cos \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{4}bm+\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\cos \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{4}b+{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\cos \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}bm-3\,{\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\cos \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}b-{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\cos \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}b-3\, \left ( \cos \left ( x \right ) \right ) ^{m}y\sin \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) }{-\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sin \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{4}bm+\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sin \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{4}b+{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\sin \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}bm-3\,{\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\sin \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}b-{\mbox{$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\sin \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}b+3\, \left ( \cos \left ( x \right ) \right ) ^{m}y\cos \left ( b\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ){\mbox{$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) }} \right ) \] Mathematica answer is simpler

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53.4 problem number 4

problem number 488

Added January 20, 2019.

Problem 2.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+m y \cot x+b^2 \sin ^m x \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (b^2 \sin ^m(x)+m y \cot (x)+y^2\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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53.5 problem number 5

problem number 489

Added January 20, 2019.

Problem 2.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 8\,{\frac{\sqrt{-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \left ( y\sin \left ( 2\,\lambda \,x \right ) -2\,\cos \left ( 2\,\lambda \,x \right ) \lambda \right ) }{4\,\sin \left ( \lambda \,x \right ) y \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}-8\,y\sin \left ( 2\,\lambda \,x \right ) \ln \left ( \cos \left ( \lambda \,x \right ) +1/2\,\sqrt{-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \right ) \sqrt{-2+2\,\cos \left ( 2\,\lambda \,x \right ) }+16\,\ln \left ( \cos \left ( \lambda \,x \right ) +1/2\,\sqrt{-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \right ) \sqrt{-2+2\,\cos \left ( 2\,\lambda \,x \right ) }\cos \left ( 2\,\lambda \,x \right ) \lambda -\sin \left ( \lambda \,x \right ) y+14\,\lambda \,\cos \left ( \lambda \,x \right ) -2\,y\sin \left ( 5\,\lambda \,x \right ) +y\sin \left ( 7\,\lambda \,x \right ) -2\,\cos \left ( 5\,\lambda \,x \right ) \lambda +2\,\cos \left ( 7\,\lambda \,x \right ) \lambda -14\,\cos \left ( 3\,\lambda \,x \right ) \lambda }} \right ) \]

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53.6 problem number 6

problem number 490

Added January 20, 2019.

Problem 2.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{(2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{a}^{2}-3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a\lambda -2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) ya+3\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) y\lambda -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}ab+3\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}b\lambda ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac{a}{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac{b}{\lambda }}} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac{a-\lambda }{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac{b-\lambda }{\lambda }}} \left ( 4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(2,-{\frac{-2\,\lambda +a+b}{\lambda }};\,-1/2\,{\frac{-5\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}a\lambda +4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(2,-{\frac{-2\,\lambda +a+b}{\lambda }};\,-1/2\,{\frac{-5\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}b\lambda -4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(2,-{\frac{-2\,\lambda +a+b}{\lambda }};\,-1/2\,{\frac{-5\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}{\lambda }^{2}-2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}{a}^{2}+5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}a\lambda -3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}{\lambda }^{2}-2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) y{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}a+3\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) y{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}\lambda +2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}ab-2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}a\lambda -3\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}b\lambda +3\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\mbox{$_2$F$_1$}(1,-{\frac{a+b-\lambda }{\lambda }};\,-1/2\,{\frac{-3\,\lambda +2\,a}{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}{\lambda }^{2} \right ) ^{-1}} \right ) \]

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53.7 problem number 7

problem number 491

Added January 20, 2019.

Problem 2.6.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \cos ^n(\lambda x) y-a \cos ^{n-1}(\lambda x) \right ) w_y = 0 \]

Mathematica

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

Maple

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

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53.8 problem number 8

problem number 492

Added January 20, 2019.

Problem 2.6.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{(\cos \left ( \lambda \,x \right ) y-1){{\rm e}^{{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) y{{\rm e}^{{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\Ei \left ( 1,{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) a-{{\rm e}^{{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\Ei \left ( 1,{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) a-\lambda \,y \right ) ^{-1}} \right ) \]

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53.9 problem number 9

problem number 493

Added January 20, 2019.

Problem 2.6.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{(\cos \left ( \lambda \,x \right ) y-1){{\rm e}^{{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) y{{\rm e}^{{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\Ei \left ( 1,{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) a-{{\rm e}^{{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\Ei \left ( 1,{\frac{\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) a-\lambda \,y \right ) ^{-1}} \right ) \]

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53.10 problem number 10

problem number 494

Added January 20, 2019.

Problem 2.6.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( A e^{\lambda x} \cos (a y) + B e^{\mu x} \sin (a y) + A e^{\lambda x} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (A e^{\lambda x} \cos (a y)+B e^{\mu x} \sin (a y)+A e^{\lambda x}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{1}{a \left ( \lambda -\mu \right ) \left ( \cos \left ( 1/2\,ay \right ) \right ) ^{3}} \left ( \int \!{{\rm e}^{-{\frac{B{{\rm e}^{\mu \,x}}a-\lambda \,x\mu }{\mu }}}}\,{\rm d}xa\cos \left ( ay \right ) A\cos \left ( 1/2\,ay \right ) -{{\rm e}^{-{\frac{B{{\rm e}^{\mu \,x}}a}{\mu }}}}\sin \left ( ay \right ) \cos \left ( 1/2\,ay \right ) +\int \!{{\rm e}^{-{\frac{B{{\rm e}^{\mu \,x}}a-\lambda \,x\mu }{\mu }}}}\,{\rm d}xaA\cos \left ( 1/2\,ay \right ) \right ) } \right ) \]

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53.11 problem number 11

problem number 495

Added January 20, 2019.

Problem 2.6.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \sin ^{n+1}(2 x) w_x + \left ( a y^2 \sin ^{2 n}x + b \cos ^{2 n} x \right ) w_y = 0 \]

Mathematica

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

Maple

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