6.4.7 3.2

   6.4.7.1 [1070] Problem 1
   6.4.7.2 [1071] Problem 2
   6.4.7.3 [1072] Problem 3
   6.4.7.4 [1073] Problem 4
   6.4.7.5 [1074] Problem 5
   6.4.7.6 [1075] Problem 6
   6.4.7.7 [1076] Problem 7

6.4.7.1 [1070] Problem 1

problem number 1070

Added Feb. 23, 2019.

Problem Chapter 4.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c y e^{\lambda x} + k x e^{\mu y} ) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \exp \left (-\frac{b c e^{\lambda x}}{a^2 \lambda ^2}-\frac{a k e^{\mu y}}{b^2 \mu ^2}+\frac{c y e^{\lambda x}}{a \lambda }+\frac{k x e^{\mu y}}{b \mu }\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{{\frac{{b}^{2}c{\mu }^{2} \left ( y\lambda \,a-b \right ){{\rm e}^{\lambda \,x}}-{a}^{2}{{\rm e}^{\mu \,y}}k{\lambda }^{2} \left ( -b\mu \,x+a \right ) }{{a}^{2}{\lambda }^{2}{\mu }^{2}{b}^{2}}}}}\]

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6.4.7.2 [1071] Problem 2

problem number 1071

Added Feb. 23, 2019.

Problem Chapter 4.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a x e^{\lambda x+\mu y} w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{y}{x}\right ) e^{\frac{a x e^{\lambda x+\mu y}}{\lambda x+\mu y}}\right \}\right \}\]

Maple

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

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6.4.7.3 [1072] Problem 3

problem number 1072

Added Feb. 23, 2019.

Problem Chapter 4.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = (a y e^{\lambda x}+ b x e^{\mu y}) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{y}{x}\right ) e^{\frac{a y e^{\lambda x}}{\lambda x}+\frac{b x e^{\mu y}}{\mu y}}\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{y}{x}} \right ){{\rm e}^{{\frac{x}{\lambda \,\mu \,y} \left ({\frac{a{{\rm e}^{\lambda \,x}}{y}^{2}\mu }{{x}^{2}}}+{{\rm e}^{\mu \,y}}b\lambda \right ) }}}\]

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6.4.7.4 [1073] Problem 4

problem number 1073

Added Feb. 23, 2019.

Problem Chapter 4.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^k w_x + b e^{\lambda y} w_y = (c x^n+s) w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{x}^{1-k}\lambda \,b-{{\rm e}^{-y\lambda }}a \left ( k-1 \right ) }{b\lambda \, \left ( k-1 \right ) }} \right ){{\rm e}^{-{\frac{{x}^{1-k} \left ( c \left ( k-1 \right ){x}^{n}+s \left ( -n-1+k \right ) \right ) }{a \left ( k-1 \right ) \left ( -n-1+k \right ) }}}}\]

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6.4.7.5 [1074] Problem 5

problem number 1074

Added Feb. 23, 2019.

Problem Chapter 4.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a y^k w_x + b e^{\lambda x} w_y = (c e^{\mu x}+s) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{y^{k+1}}{k+1}-\frac{b e^{\lambda x}}{a \lambda }\right ) \exp \left (-\frac{y^{k+1} \left (\left (y^{k+1}\right )^{\frac{1}{k+1}}\right )^{-k} \left (c \lambda e^{\mu x} \, _2F_1\left (1,\frac{\lambda +k \mu +\mu }{k \lambda +\lambda };\frac{\lambda +\mu }{\lambda };\frac{b e^{\lambda x} (k+1)}{b e^{\lambda x} (k+1)-a \lambda y^{k+1}}\right )-(k+1) \mu s \, _2F_1\left (1,\frac{1}{k+1};\frac{k+2}{k+1};\frac{a \lambda y^{k+1}}{a \lambda y^{k+1}-b e^{\lambda x} (k+1)}\right )\right )}{\mu \left (b (k+1) e^{\lambda x}-a \lambda y^{k+1}\right )}\right )\right \}\right \}\]

Maple

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

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6.4.7.6 [1075] Problem 6

problem number 1075

Added Feb. 23, 2019.

Problem Chapter 4.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda x} w_x + b y^k w_y = (c x^n+s) w \]

Mathematica

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

Maple

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

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6.4.7.7 [1076] Problem 7

problem number 1076

Added Feb. 23, 2019.

Problem Chapter 4.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a e^{\lambda y} w_x + b x^k w_y = (c e^{\mu x}+s) w \]

Mathematica

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

Maple

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

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