108 HFOPDE, chapter 4.5.2

 108.1 Problem 1
 108.2 Problem 2
 108.3 Problem 3
 108.4 Problem 4
 108.5 Problem 5
 108.6 Problem 6

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108.1 Problem 1

problem number 901

Added Feb. 25, 2019.

Problem Chapter 4.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){{\rm e}^{\int ^{x}\!{\frac{c{{\it \_a}}^{n}}{a}}+{\frac{s}{a} \left ( \ln \left ( \gamma \right ) +\ln \left ({\frac{{\it \_a}\,b+ay-bx}{a}} \right ) \right ) ^{k}}{d{\it \_a}}}} \]

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108.2 Problem 2

problem number 902

Added Feb. 25, 2019.

Problem Chapter 4.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = (b y^2+c x^n y+ s \ln ^k(\lambda x)) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\frac{s \log ^k(\lambda x) (-\log (\lambda x))^{-k} \text{Gamma}(k+1,-\log (\lambda x))}{\lambda }+\frac{1}{3} a^2 b x^3+a b x^2 (y-a x)+b x (y-a x)^2+x^n \left (\frac{a c x^2}{n+2}+\frac{c x (y-a x)}{n+1}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ){{\rm e}^{\int ^{x}\!{a}^{2}b{{\it \_a}}^{2}+a{{\it \_a}}^{n+1}c+2\, \left ( -ax+y \right ) ab{\it \_a}+{{\it \_a}}^{n} \left ( -ax+y \right ) c+ \left ( -ax+y \right ) ^{2}b+s \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}{d{\it \_a}}}} \]

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108.3 Problem 3

problem number 903

Added March 9, 2019.

Problem Chapter 4.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ){{\rm e}^{\int ^{x}\!b \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k} \left ( \ln \left ( \beta \, \left ({\it \_a}\,a-ax+y \right ) \right ) \right ) ^{n}{d{\it \_a}}}} \]

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108.4 Problem 4

problem number 904

Added March 9, 2019.

Problem Chapter 4.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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108.5 Problem 5

problem number 905

Added March 9, 2019.

Problem Chapter 4.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ) \left ({x}^{{\frac{b}{a}}} \right ) ^{{\frac{m{x}^{k}}{ka}}} \left ( y{x}^{-{\frac{b}{a}}} \right ) ^{{\frac{m{x}^{k}}{ka}}}{x}^{{\frac{{x}^{k}n}{ka}}}{{\rm e}^{1/2\,{\frac{{x}^{k}}{{a}^{2}{k}^{2}} \left ( i\pi \,akm{\it csgn} \left ( iy{x}^{-{\frac{b}{a}}} \right ) \left ({\it csgn} \left ( iy \right ) \right ) ^{2}-i\pi \,akm{\it csgn} \left ( iy{x}^{-{\frac{b}{a}}} \right ){\it csgn} \left ( i{x}^{{\frac{b}{a}}} \right ){\it csgn} \left ( iy \right ) -i\pi \,akm \left ({\it csgn} \left ( iy \right ) \right ) ^{3}+i\pi \,akm{\it csgn} \left ( i{x}^{{\frac{b}{a}}} \right ) \left ({\it csgn} \left ( iy \right ) \right ) ^{2}-2\,na-2\,mb \right ) }}} \]

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108.6 Problem 6

problem number 906

Added March 9, 2019.

Problem Chapter 4.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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