6.4.14 5.2

   6.4.14.1 [1109] Problem 1
   6.4.14.2 [1110] Problem 2
   6.4.14.3 [1111] Problem 3
   6.4.14.4 [1112] Problem 4
   6.4.14.5 [1113] Problem 5
   6.4.14.6 [1114] Problem 6

6.4.14.1 [1109] Problem 1

problem number 1109

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

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( c{{\it \_a}}^{n}+s \left ( -3\,\ln \left ( 2 \right ) +\ln \left ({\frac{ya-b \left ( x-{\it \_a} \right ) }{a}} \right ) \right ) ^{k} \right ) }{d{\it \_a}}}}\]

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6.4.14.2 [1110] Problem 2

problem number 1110

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

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

Maple

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

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6.4.14.3 [1111] Problem 3

problem number 1111

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

\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xb \log ^k(\lambda K[1]) \log ^n(\beta (y+a (K[1]-x)))dK[1]\right )\right \}\right \}\]

Maple

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

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6.4.14.4 [1112] Problem 4

problem number 1112

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

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

Maple

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

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6.4.14.5 [1113] Problem 5

problem number 1113

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

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

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6.4.14.6 [1114] Problem 6

problem number 1114

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac{b x^{1-k}}{a (k-1)}-\frac{y^{1-n}}{n-1}\right ) \exp \left (\int _1^x\frac{K[1]^{-k} \left (c \log ^m(\lambda K[1])+s \log ^t\left (\beta \left (\frac{a (k-1) x^k y^n K[1]^k}{a (k-1) x^k y K[1]^k-b (n-1) y^n \left (x K[1]^k-x^k K[1]\right )}\right )^{\frac{1}{n-1}}\right )\right )}{a}dK[1]\right )\right \}\right \}\]

Maple

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

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