101 HFOPDE, chapter 4.3.2

 101.1 Problem 1
 101.2 Problem 2
 101.3 Problem 3
 101.4 Problem 4
 101.5 Problem 5
 101.6 Problem 6
 101.7 Problem 7

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101.1 Problem 1

problem number 862

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

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){{\rm e}^{-{\frac{1}{{\lambda }^{2}{a}^{2}{\mu }^{2}{b}^{2}} \left ( -xk{{\rm e}^{{\frac{\mu \, \left ( ay-bx \right ) }{a}}+{\frac{\mu \,bx}{a}}}}\mu \,b{a}^{2}{\lambda }^{2}- \left ( ay-bx \right ) c{{\rm e}^{\lambda \,x}}\lambda \,{\mu }^{2}{b}^{2}-{{\rm e}^{\lambda \,x}}{b}^{3}c\lambda \,{\mu }^{2}x+{a}^{3}k{{\rm e}^{{\frac{\mu \, \left ( ay-bx \right ) }{a}}+{\frac{\mu \,bx}{a}}}}{\lambda }^{2}+{{\rm e}^{\lambda \,x}}{b}^{3}c{\mu }^{2} \right ) }}} \]

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101.2 Problem 2

problem number 863

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

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

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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101.3 Problem 3

problem number 864

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

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

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101.4 Problem 4

problem number 865

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

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

Maple

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

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101.5 Problem 5

problem number 866

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

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{a (k+1) \lambda }\right ) \exp \left (\frac{c e^{\mu x} \left (\frac{b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}+1\right )^{\frac{k}{k+1}} \left (\left (\frac{(k+1) \left (\frac{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac{1}{k+1}}\right )^{-k} \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{\mu }{\lambda },\frac{\lambda +\mu }{\lambda },-\frac{b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}\right )}{a \mu }-\frac{(k+1) s \left (\left (\frac{(k+1) \left (\frac{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac{1}{k+1}}\right )^{-k} \left (\frac{e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}+1\right )^{\frac{k}{k+1}} \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{k}{k+1},\frac{k}{k+1}+1,-\frac{e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}\right )}{a k \lambda }\right )\right \}\right \} \]

Maple

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

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101.6 Problem 6

problem number 867

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

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

Maple

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

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101.7 Problem 7

problem number 868

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

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

Maple

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