81 HFOPDE, chapter 3.5.3

 81.1 Problem 4
 81.2 Problem 5
 81.3 Problem 6

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81.1 Problem 4

problem number 726

Added Feb. 11, 2019.

Problem Chapter 3.5.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+b x^n) w_y = c ln^k(\lambda x) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{(-\log (\lambda x))^{-k} \left (\lambda (-\log (\lambda x))^k c_1\left (a^{-n-1} e^{-a x} \left (b e^{a x} \text{Gamma}(n+1,a x)+y a^{n+1}\right )\right )+c \log ^k(\lambda x) \text{Gamma}(k+1,-\log (\lambda x))\right )}{\lambda }\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int \!c \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+{\it \_F1} \left ( -{\frac{{{\rm e}^{-ax}} \left ( \left ( ax \right ) ^{-n/2} \WhittakerM \left ( n/2,n/2+1/2,ax \right ){{\rm e}^{1/2\,ax}}{x}^{n}b-ayn-ay \right ) }{a \left ( n+1 \right ) }} \right ) \] Result has unresolved integrals

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81.2 Problem 5

problem number 727

Added Feb. 11, 2019.

Problem Chapter 3.5.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = x^k ( n \ln x+ m \ln y) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a^2 k^2 c_1\left (y x^{-\frac{b}{a}}\right )+a k m x^k \log (y)-a n x^k+a k n x^k \log (x)-b m x^k}{a^2 k^2}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =1/2\,{\frac{1}{{a}^{2}{k}^{2}} \left ( i\pi \,{\it csgn} \left ( iy{x}^{-{\frac{b}{a}}} \right ) \left ({\it csgn} \left ( iy \right ) \right ) ^{2}{x}^{k}akm-i\pi \,{\it csgn} \left ( iy{x}^{-{\frac{b}{a}}} \right ){\it csgn} \left ( iy \right ){\it csgn} \left ( i{x}^{{\frac{b}{a}}} \right ){x}^{k}akm-i\pi \, \left ({\it csgn} \left ( iy \right ) \right ) ^{3}{x}^{k}akm+i\pi \, \left ({\it csgn} \left ( iy \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{{\frac{b}{a}}} \right ){x}^{k}akm+2\,\ln \left ( x \right ){x}^{k}akn+2\,\ln \left ( y{x}^{-{\frac{b}{a}}} \right ){x}^{k}akm+2\,{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ){a}^{2}{k}^{2}+2\,m{x}^{k}\ln \left ({x}^{{\frac{b}{a}}} \right ) ak-2\,an{x}^{k}-2\,{x}^{k}bm \right ) } \]

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81.3 Problem 6

problem number 728

Added Feb. 11, 2019.

Problem Chapter 3.5.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^k w_x + b y^n w_y = c \ln ^m(\lambda x)+s \ln ^l(\beta y) \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{{{\it \_a}}^{-k}}{a} \left ( c\ln \left ( \lambda \,{\it \_a} \right ) +s \left ( \ln \left ( \beta \, \left ({\frac{b \left ( n-1 \right ){{\it \_a}}^{-k+1}-{x}^{-k+1}b \left ( n-1 \right ) +{y}^{1-n}a \left ( k-1 \right ) }{a \left ( k-1 \right ) }} \right ) ^{- \left ( n-1 \right ) ^{-1}} \right ) \right ) ^{l} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{-{x}^{-k+1}b \left ( n-1 \right ) +{y}^{1-n}a \left ( k-1 \right ) }{a \left ( k-1 \right ) }} \right ) \]