72 HFOPDE, chapter 3.2.2

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72.1 Problem 1

problem number 662

Added Feb. 9, 2019.

Problem Chapter 3.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c x^2+d y^2+ k x y+n \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^2 + d*y^2 + k*x*y + n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {6 a^3 c_1\left (\frac {a y-b x}{a}\right )+2 a^2 c x^3+6 a^2 d x y^2+3 a^2 k x^2 y+6 a^2 n x-6 a b d x^2 y-a b k x^3+2 b^2 d x^3}{6 a^3}\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := a* diff(w(x,y),x)+b*diff(w(x,y),y) = c*x^2+d*y^2+k*x*y+n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =1/6\,{\frac { \left ( 2\,c{a}^{2}-abk+2\,{b}^{2}d \right ) {x}^{3}}{{a}^{3}}}+1/6\,{\frac { \left ( 3\,k{a}^{2}-6\,abd \right ) y{x}^{2}}{{a}^{3}}}+ \left ( {\frac {d{y}^{2}}{a}}+{\frac {n}{a}} \right ) x+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

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72.2 Problem 2

problem number 663

Added Feb. 9, 2019.

Problem Chapter 3.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c x^2+d y^2+ k x y+n \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^2 + d*y^2 + k*x*y + n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := a*x*diff(w(x,y),x)+b*y*diff(w(x,y),y) = c*x^2+d*y^2+k*x*y+n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {kxy}{a+b}}+1/2\,{\frac {c{x}^{2}}{a}}+{\frac {n\ln \left ( x \right ) }{a}}+1/2\,{\frac {d{y}^{2}}{b}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]

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72.3 Problem 3

problem number 664

Added Feb. 9, 2019.

Problem Chapter 3.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == c*x*y + d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := a*y*diff(w(x,y),x)+b*x*diff(w(x,y),y) = c*x*y+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =1/2\,{\frac {c{x}^{2}}{a}}+1/2\,{\frac {1}{a\sqrt {ab}} \left ( 2\,{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) a\sqrt {ab}+2\,d\ln \left ( {\frac {axb}{\sqrt {ab}}}+\sqrt {ab{x}^{2}+ \left ( {y}^{2}a-b{x}^{2} \right ) a} \right ) a \right ) } \]

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72.4 Problem 4

problem number 665

Added Feb. 9, 2019.

Problem Chapter 3.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^2 w_x + b y^2 w_y = c x^2+d y^2+ k x y+ n x+ m y+s \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*x^2*D[w[x, y], x] + b*y^2*D[w[x, y], y] == c*x^2 + d*y^2 + k*x*y + n*x + m*y + s; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := a*x^2*diff(w(x,y),x)+b*y^2*diff(w(x,y),y) =c*x^2+d*y^2+ k*x*y+ n*x+ m*y+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {cx}{a}}+{\frac {adxy}{-ax+by} \left ( -{\frac {-ax+by}{y}}+b \right ) ^{-1}}-{\frac {m}{b}\ln \left ( -{\frac {-ax+by}{y}}+b \right ) }-{\frac {kxy}{-ax+by}\ln \left ( -{\frac {-ax+by}{y}}+b \right ) }-{\frac {s}{ax}}+{\frac {m\ln \left ( x \right ) }{b}}+{\frac {n\ln \left ( x \right ) }{a}}+{\it \_F1} \left ( -{\frac {-ax+by}{yax}} \right ) \]

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72.5 Problem 5

problem number 666

Added Feb. 9, 2019.

Problem Chapter 3.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + a x y w_y = b y^2 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = x^2*D[w[x, y], x] + a*x*y*D[w[x, y], y] == b*y^2; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := x^2*diff(w(x,y),x)+a*x*y*diff(w(x,y),y) =b*y^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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72.6 Problem 6

problem number 667

Added Feb. 9, 2019.

Problem Chapter 3.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^2 w_x + b x^2 w_y = c x^2+d \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*y^2*D[w[x, y], x] + b*x^2*D[w[x, y], y] == c*x^2 + d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {b d x \left (\frac {a y^3}{a y^3-b x^3}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a y^3-b x^3}\right )+\sqrt [3]{a} b \left (a y^3\right )^{2/3} c_1\left (\frac {a y^3-b x^3}{3 a}\right )+a c y^3}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}\right \},\left \{w(x,y)\to \frac {-\sqrt [3]{-1} b d x \left (\frac {a y^3}{a y^3-b x^3}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a y^3-b x^3}\right )+\sqrt [3]{a} b \left (a y^3\right )^{2/3} c_1\left (\frac {a y^3-b x^3}{3 a}\right )-\sqrt [3]{-1} a c y^3}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}\right \},\left \{w(x,y)\to \frac {(-1)^{2/3} b d x \left (\frac {a y^3}{a y^3-b x^3}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a y^3-b x^3}\right )+\sqrt [3]{a} b \left (a y^3\right )^{2/3} c_1\left (\frac {a y^3-b x^3}{3 a}\right )+(-1)^{2/3} a c y^3}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := a*y^2*diff(w(x,y),x)+b*x^2*diff(w(x,y),y) =c*x^2+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac { \left ( {{\it \_a}}^{2}c+d \right ) a}{ \left ( \left ( {{\it \_a}}^{3}b+\RootOf \left ( ya-\sqrt [3]{{a}^{2}b{x}^{3}+{a}^{3}{\it \_Z}} \right ) a \right ) {a}^{2} \right ) ^{2/3}}}{d{\it \_a}}+{\it \_F1} \left ( \RootOf \left ( ya-\sqrt [3]{{a}^{2}b{x}^{3}+{a}^{3}{\it \_Z}} \right ) \right ) \] Contains unresolved integral with RootOf

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72.7 Problem 7

problem number 668

Added Feb. 9, 2019.

Problem Chapter 3.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^2 w_x + b x y w_y = c x^2+d y^2 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 
 pde = a*y^2*D[w[x, y], x] + b*x*y*D[w[x, y], y] == c*x^2 + d*y^2; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := a*y^2*diff(w(x,y),x)+b*x*y*diff(w(x,y),y) =c*x^2+d*y^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {c{x}^{2}}{\sqrt { \left ( {y}^{2}a-b{x}^{2} \right ) b}}\arctan \left ( {\frac {bx}{\sqrt { \left ( {y}^{2}a-b{x}^{2} \right ) b}}} \right ) }+ \left ( {\frac {c}{b}}+{\frac {d}{a}} \right ) x-{\frac {c{y}^{2}a}{b\sqrt { \left ( {y}^{2}a-b{x}^{2} \right ) b}}\arctan \left ( {\frac {bx}{\sqrt { \left ( {y}^{2}a-b{x}^{2} \right ) b}}} \right ) }+{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \]