71 HFOPDE, chapter 3.2.4

 71.1 Problem 1
 71.2 Problem 2
 71.3 Problem 3
 71.4 Problem 4
 71.5 Problem 5
 71.6 Problem 6
 71.7 Problem 7
 71.8 Problem 8
 71.9 Problem 9
 71.10 Problem 10
 71.11 Problem 11

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71.1 Problem 1

problem number 659

Added Feb. 9, 2019.

Problem Chapter 3.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x +b w_y = c x^n + d y^m \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{d}{b \left ( m+1 \right ) } \left ({\frac{bx}{a}}+{\frac{ay-bx}{a}} \right ) ^{m+1}}+{\frac{{x}^{n+1}c}{a \left ( n+1 \right ) }}+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

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71.2 Problem 2

problem number 660

Added Feb. 9, 2019.

Problem Chapter 3.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x +b w_y = c x^n y \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a^2 n^2 c_1\left (\frac{a y-b x}{a}\right )+3 a^2 n c_1\left (\frac{a y-b x}{a}\right )+2 a^2 c_1\left (\frac{a y-b x}{a}\right )+2 a c y x^{n+1}+a c n y x^{n+1}-b c x^{n+2}}{a^2 (n+1) (n+2)}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =-{\frac{{x}^{n+1}bcx}{ \left ( 2+n \right ) \left ( n+1 \right ){a}^{2}}}+{\frac{{x}^{n+1} \left ( an+2\,a \right ) cy}{ \left ( 2+n \right ) \left ( n+1 \right ){a}^{2}}}+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

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71.3 Problem 3

problem number 661

Added Feb. 9, 2019.

Problem Chapter 3.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x +y w_y = a(x^2+y^2)^k \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a \left (x^2 \left (\frac{y^2}{x^2}+1\right )\right )^k+2 k c_1\left (\frac{y}{x}\right )}{2 k}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =1/2\,{\frac{1}{k} \left ( a \left ({x}^{2} \left ({\frac{{y}^{2}}{{x}^{2}}}+1 \right ) \right ) ^{k}+2\,{\it \_F1} \left ({\frac{y}{x}} \right ) k \right ) } \]

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71.4 Problem 4

problem number 662

Added Feb. 9, 2019.

Problem Chapter 3.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x +b y w_y = c x^n y^m \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{c{x}^{n}{y}^{m}}{an+mb}}+{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ) \]

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71.5 Problem 5

problem number 663

Added Feb. 9, 2019.

Problem Chapter 3.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x +b y w_y = c x^n + d y^m \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{a{\it \_a}} \left ( c{{\it \_a}}^{n}+d \left ( y{x}^{-{\frac{b}{a}}}{{\it \_a}}^{{\frac{b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ) \] Result has unresolved integral

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71.6 Problem 6

problem number 664

Added Feb. 9, 2019.

Problem Chapter 3.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{knm} \left ({\it \_F1} \left ( y{x}^{-{\frac{n}{m}}} \right ) knm+ \left ( a{x}^{n}+b{y}^{m} \right ) ^{k} \right ) } \] Result has unresolved integral

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71.7 Problem 7

problem number 665

Added Feb. 9, 2019.

Problem Chapter 3.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x +b y^m w_y = c x^k+ d y^s \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{{x}^{k-n+1}c}{a \left ( k-n+1 \right ) }}-{\frac{d}{ \left ( mn-ns-m-n+s+1 \right ) n}{a}^{{\frac{s}{m-1}}-1} \left ({x}^{1-n}nm-{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{n-1}}-{x}^{1-n}n+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{n-1}} \right ) \left ( n-1 \right ) ^{{\frac{s}{m-1}}} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) ^{-{\frac{s}{m-1}}}{{\rm e}^{{\frac{i/2s\pi }{m-1} \left ({\it csgn} \left ({\frac{i}{n-1} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ) \right ) ^{3}}-{\frac{i/2s\pi }{m-1} \left ({\it csgn} \left ({\frac{i}{n-1} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{n-1}} \right ) }-{\frac{i/2s\pi }{m-1} \left ({\it csgn} \left ({\frac{i}{n-1} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ) \right ) ^{2}{\it csgn} \left ( i \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) \right ) }+{\frac{i/2s\pi }{m-1}{\it csgn} \left ({\frac{i}{n-1} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ){\it csgn} \left ({\frac{i}{n-1}} \right ){\it csgn} \left ( i \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) \right ) }-{\frac{i/2s\pi }{m-1}{\it csgn} \left ({\frac{i}{n-1} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ) \left ({\it csgn} \left ({\frac{i}{ \left ( n-1 \right ) a} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ) \right ) ^{2}}+{\frac{i/2s\pi }{m-1}{\it csgn} \left ({\frac{i}{n-1} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ){\it csgn} \left ({\frac{i}{ \left ( n-1 \right ) a} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ){\it csgn} \left ({\frac{i}{a}} \right ) }+{\frac{i/2s\pi }{m-1} \left ({\it csgn} \left ({\frac{i}{ \left ( n-1 \right ) a} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ) \right ) ^{3}}-{\frac{i/2s\pi }{m-1} \left ({\it csgn} \left ({\frac{i}{ \left ( n-1 \right ) a} \left ( \left ( -{\frac{ \left ({x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a \right ) n}{ \left ( n-1 \right ) a}}+{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) a+{x}^{1-n}nm-{x}^{1-n}n \right ) } \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{a}} \right ) }}}}+{\it \_F1} \left ( -{\frac{{x}^{1-n}nm-{y}^{-m+1}an-{x}^{1-n}n+{y}^{-m+1}a}{ \left ( n-1 \right ) a}} \right ) \] Result has unresolved integral

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71.8 Problem 8

problem number 666

Added Feb. 9, 2019.

Problem Chapter 3.2.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x +b x^m y w_y = c x^k y^s + d \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{a} \left ({{\it \_a}}^{k-n}c \left ( y{{\rm e}^{-{\frac{n{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}+{\frac{n{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \right ) ^{s}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac{n{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \right ) \] Result has unresolved integral

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71.9 Problem 9

problem number 667

Added Feb. 9, 2019.

Problem Chapter 3.2.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x +(b x^m y + c x^k) w_y = s x^p y^q + d \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{a} \left ({{\it \_a}}^{-n+p}s \left ( y{{\rm e}^{-{\frac{n{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}+{\frac{n{{\it \_a}}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac{n{x}^{m-n+1}}{a \left ( m-n+1 \right ) }}}} \right ) \]

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71.10 Problem 10

problem number 668

Added Feb. 9, 2019.

Problem Chapter 3.2.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x +(b x^m y^k + c x^l y) w_y = s x^p y^q + d \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{a} \left ({{\it \_a}}^{-n+p}s \left ( \left ( -{\frac{1}{a} \left ({\frac{bk}{-n+l+1} \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac{m-n+1}{-n+l+1}}} \left ({\frac{ \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l}a}{ \left ( m-n+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ){l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ){n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ({\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}},1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac{ \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( m-n+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) }-{\frac{b}{-n+l+1} \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac{m-n+1}{-n+l+1}}} \left ({\frac{ \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l}a}{ \left ( m-n+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ){l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ){n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ({\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}},1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac{ \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( m-n+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{{{\it \_a}}^{-n+l+1}c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) }-bk\int \!{{\rm e}^{{\frac{{x}^{-n+l+1}c \left ( k-1 \right ) }{ \left ( -n+l+1 \right ) a}}}}{x}^{m-n}\,{\rm d}x-{{y}^{ \left ( -n+l+1 \right ) ^{-1}}a{{\rm e}^{{\frac{c{x}^{-n+l+1}k}{ \left ( -n+l+1 \right ) a}}}}{y}^{{\frac{kn}{-n+l+1}}}{y}^{{\frac{l}{-n+l+1}}} \left ({y}^{{\frac{kl}{-n+l+1}}} \right ) ^{-1} \left ({y}^{{\frac{k}{-n+l+1}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{c{x}^{-n+l+1}}{ \left ( -n+l+1 \right ) a}}}} \right ) ^{-1} \left ({y}^{{\frac{n}{-n+l+1}}} \right ) ^{-1}}+b\int \!{{\rm e}^{{\frac{{x}^{-n+l+1}c \left ( k-1 \right ) }{ \left ( -n+l+1 \right ) a}}}}{x}^{m-n}\,{\rm d}x \right ) } \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{{\frac{{{\it \_a}}^{-n+l+1}c}{ \left ( -n+l+1 \right ) a}}}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{1}{a} \left ({\frac{bk}{-n+l+1} \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac{m-n+1}{-n+l+1}}} \left ({\frac{ \left ( -n+l+1 \right ) ^{2}{x}^{m-l}a}{ \left ( m-n+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{c{x}^{-n+l+1} \left ( -k+1 \right ){l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ){n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ({\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}},1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac{ \left ( -n+l+1 \right ) ^{2}{x}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( m-n+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) }+{{y}^{ \left ( -n+l+1 \right ) ^{-1}}a{{\rm e}^{{\frac{c{x}^{-n+l+1}k}{ \left ( -n+l+1 \right ) a}}}}{y}^{{\frac{kn}{-n+l+1}}}{y}^{{\frac{l}{-n+l+1}}} \left ({y}^{{\frac{kl}{-n+l+1}}} \right ) ^{-1} \left ({y}^{{\frac{k}{-n+l+1}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{c{x}^{-n+l+1}}{ \left ( -n+l+1 \right ) a}}}} \right ) ^{-1} \left ({y}^{{\frac{n}{-n+l+1}}} \right ) ^{-1}}-{\frac{b}{-n+l+1} \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac{m-n+1}{-n+l+1}}} \left ({\frac{ \left ( -n+l+1 \right ) ^{2}{x}^{m-l}a}{ \left ( m-n+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{c{x}^{-n+l+1} \left ( -k+1 \right ){l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ){n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ({\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}},1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac{ \left ( -n+l+1 \right ) ^{2}{x}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( m-n+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( -k+1 \right ) } \left ({\frac{c \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac{m-n+1}{-n+l+1}}} \left ({\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ({\frac{m-n+1}{-n+l+1}}-1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac{l+m-2\,n+2}{-n+l+1}}+1/2,{\frac{c{x}^{-n+l+1} \left ( -k+1 \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) } \right ) } \right ) \]

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71.11 Problem 11

problem number 669

Added Feb. 9, 2019.

Problem Chapter 3.2.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b x^m w_y = c x^m + d \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{\left (\left (\frac{a (m+1)}{(k+1) \left (b x^{m+1}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{k+1}\right )}\right )^{-\frac{1}{k+1}}\right )^{-k} \left (a b c_1\left (\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{a (k+1) (m+1)}\right ) \left (\left (\frac{a (m+1)}{(k+1) \left (b x^{m+1}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{k+1}\right )}\right )^{-\frac{1}{k+1}}\right )^k+a b m c_1\left (\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{a (k+1) (m+1)}\right ) \left (\left (\frac{a (m+1)}{(k+1) \left (b x^{m+1}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{k+1}\right )}\right )^{-\frac{1}{k+1}}\right )^k-a c y^{k+1} \left (\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac{k}{k+1}}-a c m y^{k+1} \left (\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac{k}{k+1}}+b c x^{m+1} \left (\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac{k}{k+1}}+b c k x^{m+1} \left (\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac{k}{k+1}}+a c y^{k+1}+a c m y^{k+1}+b d x \left (\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac{k}{k+1}} \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{1}{m+1},1+\frac{1}{m+1},-\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}\right )+b d m x \left (\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac{k}{k+1}} \text{Hypergeometric2F1}\left (\frac{k}{k+1},\frac{1}{m+1},1+\frac{1}{m+1},-\frac{b x^{m+1}}{\frac{m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac{a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}\right )\right )}{a b (m+1)}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{c{{\it \_a}}^{m}+d}{a} \left ( \left ({\frac{1}{a \left ( n+1 \right ) } \left ({{\it \_a}}^{n+1}bk+{\frac{n \left ( -{x}^{n+1}bk+{y}^{k+1}an-{x}^{n+1}b+{y}^{k+1}a \right ) }{n+1}}+{{\it \_a}}^{n+1}b+{\frac{-{x}^{n+1}bk+{y}^{k+1}an-{x}^{n+1}b+{y}^{k+1}a}{n+1}} \right ) } \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}+{\it \_F1} \left ({\frac{-{x}^{n+1}bk+{y}^{k+1}an-{x}^{n+1}b+{y}^{k+1}a}{a \left ( n+1 \right ) }} \right ) \]