63 HFOPDE, chapter 2.8.6

 63.1 problem number 1
 63.2 problem number 2
 63.3 problem number 3
 63.4 problem number 4
 63.5 problem number 5
 63.6 problem number 6
 63.7 problem number 7
 63.8 problem number 8
 63.9 problem number 9
 63.10 problem number 10
 63.11 problem number 11
 63.12 problem number 12

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63.1 problem number 1

problem number 580

Added Feb. 7, 2019.

Problem 2.8.6.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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63.2 problem number 2

problem number 581

Added Feb. 7, 2019.

Problem 2.8.6.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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63.3 problem number 3

problem number 582

Added Feb. 7, 2019.

Problem 2.8.6.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-f(x) \int _1^x g(K[2]) \, dK[2]+y \int _1^x g(K[2]) \, dK[2]+1}{y-f(x)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{\int \!g \left ( x \right ) \,{\rm d}xy-f \left ( x \right ) \int \!g \left ( x \right ) \,{\rm d}x+1}{y-f \left ( x \right ) }} \right ) \]

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63.4 problem number 4

problem number 583

Added Feb. 7, 2019.

Problem 2.8.6.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \frac{f'(x)}{g(x)} y^2 - \frac{g'(x)}{f(x)}\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{ \left ( f \left ( x \right ) y+g \left ( x \right ) \right ) f \left ( x \right ) } \left ( y \left ( f \left ( x \right ) \right ) ^{2}\int \!{\frac{{\frac{\rm d}{{\rm d}x}}f \left ( x \right ) }{g \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x+g \left ( x \right ) f \left ( x \right ) \int \!{\frac{{\frac{\rm d}{{\rm d}x}}f \left ( x \right ) }{g \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x+1 \right ) } \right ) \]

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63.5 problem number 5

problem number 584

Added Feb. 7, 2019.

Problem 2.8.6.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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63.6 problem number 6

problem number 585

Added Feb. 7, 2019.

Problem 2.8.6.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2 - \frac{f''(x)}{f(x)} \right ) w_y = 0 \]

Mathematica

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

Maple

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

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63.7 problem number 7

problem number 586

Added Feb. 7, 2019.

Problem 2.8.6.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ g(x) w_x + \left ( a f(x) g(x) y^3 + (b f(x) g^3(x) + g'(x)) y+ c f(x) g^4(x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^3 f(x) g(x)+y \left (b f(x) g(x)^3+g'(x)\right )+c f(x) g(x)^4\right )+g(x) w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({b}^{3}\sum _{{\it \_R}=\RootOf \left ( a{c}^{2}{{\it \_Z}}^{3}+{b}^{3}{\it \_Z}-{b}^{3} \right ) }{\frac{1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac{{\it \_R}\,cg \left ( x \right ) +by}{cg \left ( x \right ) }} \right ) }-b\int \! \left ( g \left ( x \right ) \right ) ^{2}f \left ( x \right ) \,{\rm d}x \right ) \]

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63.8 problem number 8

problem number 587

Added Feb. 7, 2019.

Problem 2.8.6.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^3+3 f(x) h(x) y^2+(g(x)+3 f(x) h^2(x)) y+ f(x) h^3(x) + g(x) h(x) - h'(x) \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{4 y h(x) \left (\int _1^x f(K[2]) e^{2 \text{Integrate}[g(K[1]),\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}]} \, dK[2]\right )+2 h(x)^2 \int _1^x f(K[2]) e^{2 \text{Integrate}[g(K[1]),\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}]} \, dK[2]+2 y^2 \left (\int _1^x f(K[2]) e^{2 \text{Integrate}[g(K[1]),\{K[1],1,K[2]\},\text{Assumptions}\to \text{True}]} \, dK[2]\right )+e^{2 \int _1^x g(K[1]) \, dK[1]}}{(h(x)+y)^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{2\, \left ( h \left ( x \right ) \right ) ^{2}\int \!f \left ( x \right ){{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+4\,h \left ( x \right ) y\int \!f \left ( x \right ){{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+2\,{y}^{2}\int \!f \left ( x \right ){{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{2\,\int \!g \left ( x \right ) \,{\rm d}x}}}{ \left ( h \left ( x \right ) +y \right ) ^{2}}} \right ) \]

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63.9 problem number 9

problem number 588

Added Feb. 7, 2019.

Problem 2.8.6.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \frac{g'(x)}{f^2(x) (a g(x) + b)^3} y^3 + \frac{f'(x)}{f(x)} y + f(x) g'(x) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (\frac{y^3 g'(x)}{f(x)^2 (a g(x)+b)^3}+\frac{y f'(x)}{f(x)}+f(x) g'(x)\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({a}^{3}\sum _{{\it \_R}=\RootOf \left ( -{a}^{3}{\it \_Z}+{{\it \_Z}}^{3}+{a}^{3} \right ) }{\frac{1}{-{a}^{3}+3\,{{\it \_R}}^{2}}\ln \left ({\frac{-g \left ( x \right ) f \left ( x \right ){\it \_R}\,a-f \left ( x \right ){\it \_R}\,b+ay}{ \left ( ag \left ( x \right ) +b \right ) f \left ( x \right ) }} \right ) }-\ln \left ( ag \left ( x \right ) +b \right ) \right ) \]

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63.10 problem number 10

problem number 589

Added Feb. 7, 2019.

Problem 2.8.6.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\frac{1}{{a}^{2}+2\,ab+{b}^{2}} \left ({\frac{a}{b \left ( a+b \right ) } \left ( \ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ){a}^{3}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ){a}^{2}b+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){a}^{3}-2\,\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){a}^{2}b-2\,\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){b}^{3}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ){a}^{2}b+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) a{b}^{2}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ){b}^{3} \right ) }+2\,{\frac{1}{a+b} \left ( \ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ){a}^{3}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ){a}^{2}b+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){a}^{3}-2\,\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){a}^{2}b-2\,\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){b}^{3}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ){a}^{2}b+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) a{b}^{2}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ){b}^{3} \right ) }+{\frac{b}{a \left ( a+b \right ) } \left ( \ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ){a}^{3}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ){a}^{2}b+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}f \left ( x \right ) -2\,{a}^{2}f \left ( x \right ) b-2\,af \left ( x \right ){b}^{2}-f \left ( x \right ){b}^{3}}{ag \left ( x \right ) +2\,bg \left ( x \right ) -af \left ( x \right ) -2\,f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){a}^{3}-2\,\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){a}^{2}b-2\,\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ) a{b}^{2}-\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{2}g \left ( x \right ) b-ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) -{a}^{3}f \left ( x \right ) -{a}^{2}f \left ( x \right ) b-af \left ( x \right ){b}^{2}}{ag \left ( x \right ) -bg \left ( x \right ) -af \left ( x \right ) +f \left ( x \right ) b}} \right ){b}^{3}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ){a}^{2}b+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ) a{b}^{2}+\ln \left ( -9\,{\frac{{a}^{3}y+2\,{a}^{2}by+2\,a{b}^{2}y+{b}^{3}y-{a}^{3}g \left ( x \right ) -2\,{a}^{2}g \left ( x \right ) b-2\,ag \left ( x \right ){b}^{2}-{b}^{3}g \left ( x \right ) }{2\,ag \left ( x \right ) +bg \left ( x \right ) -2\,af \left ( x \right ) -f \left ( x \right ) b}} \right ){b}^{3} \right ) }+2\,{a}^{2}\int \!h \left ( x \right ) f \left ( x \right ) g \left ( x \right ) \,{\rm d}x-{a}^{2}\int \!h \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}\,{\rm d}x-{a}^{2}\int \!h \left ( x \right ) \left ( g \left ( x \right ) \right ) ^{2}\,{\rm d}x+2\,ab\int \!h \left ( x \right ) f \left ( x \right ) g \left ( x \right ) \,{\rm d}x-ab\int \!h \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}\,{\rm d}x-ab\int \!h \left ( x \right ) \left ( g \left ( x \right ) \right ) ^{2}\,{\rm d}x+2\,{b}^{2}\int \!h \left ( x \right ) f \left ( x \right ) g \left ( x \right ) \,{\rm d}x-{b}^{2}\int \!h \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}\,{\rm d}x-{b}^{2}\int \!h \left ( x \right ) \left ( g \left ( x \right ) \right ) ^{2}\,{\rm d}x \right ) } \right ) \]

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63.11 problem number 11

problem number 590

Added Feb. 7, 2019.

Problem 2.8.6.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2 + g'(x) y+ a f(x) e^{2 g(x)} \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{y e^{-g(x)}}{\sqrt{a}}\right )-\sqrt{a} \int _1^x f(K[1]) e^{g(K[1])} \, dK[1]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \sqrt{a}\int \!f \left ( x \right ){{\rm e}^{g \left ( x \right ) }}\,{\rm d}x-\arctan \left ({\frac{{{\rm e}^{-g \left ( x \right ) }}y}{\sqrt{a}}} \right ) \right ) \]

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63.12 problem number 12

problem number 591

Added Feb. 7, 2019.

Problem 2.8.6.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f'(x) y^2+ a e^{\lambda x} f(x) y+a e^{\lambda x} \right ) w_y = 0 \]

Mathematica

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

Maple

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