61 HFOPDE, chapter 2.8.4

 61.1 problem number 1
 61.2 problem number 2
 61.3 problem number 3
 61.4 problem number 4

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61.1 problem number 1

problem number 571

Added Feb. 7, 2019.

Problem 2.8.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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61.2 problem number 2

problem number 572

Added Feb. 7, 2019.

Problem 2.8.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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61.3 problem number 3

problem number 573

Added Feb. 7, 2019.

Problem 2.8.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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61.4 problem number 4

problem number 574

Added Feb. 7, 2019.

Problem 2.8.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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