73 HFOPDE, chapter 3.2.3

____________________________________________________________________________________

73.1 Problem 1

problem number 669

Added Feb. 9, 2019.

Problem Chapter 3.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a \sqrt {x^2+y^2} \]

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*D[w[x, y], x] + y*D[w[x, y], y] == a*Sqrt[x^2 + y^2]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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*diff(w(x,y),x) + y*diff(w(x,y),y) =a*sqrt(x^2+y^2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

____________________________________________________________________________________

73.2 Problem 2

problem number 670

Added Feb. 9, 2019.

Problem Chapter 3.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*y^2 + d*x^2*y + k; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {2 a^3 c_1\left (y x^{-\frac {b}{a}}\right )+5 a^2 b c_1\left (y x^{-\frac {b}{a}}\right )+2 a^2 c x y^2+a^2 d x^2 y+2 a^2 k \log (x)+2 a b^2 c_1\left (y x^{-\frac {b}{a}}\right )+a b c x y^2+2 a b d x^2 y+5 a b k \log (x)+2 b^2 k \log (x)}{a (2 a+b) (a+2 b)}\right \}\right \} \]

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*y^2+d*x^2*y+k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

____________________________________________________________________________________

73.3 Problem 3

problem number 671

Added Feb. 9, 2019.

Problem Chapter 3.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*D[w[x, y], x] + b*x*D[w[x, y], y] == c*x*y^2 + d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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*y*diff(w(x,y),x) + b*x*diff(w(x,y),y) =c*x*y^2+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( -{\frac {cy}{a}}+{\frac {c\sqrt {ab{x}^{2}+ \left ( {y}^{2}a-b{x}^{2} \right ) a}}{{a}^{2}}} \right ) {x}^{2}+{\frac {c{y}^{3}}{b}}-2/3\,{\frac {c\sqrt {ab{x}^{2}+ \left ( {y}^{2}a-b{x}^{2} \right ) a}{y}^{2}}{ab}}-1/3\,{\frac {1}{\sqrt {ab}{a}^{2}b} \left ( -3\,{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \sqrt {ab}{a}^{2}b-3\,d\ln \left ( {\frac {axb}{\sqrt {ab}}}+\sqrt {ab{x}^{2}+ \left ( {y}^{2}a-b{x}^{2} \right ) a} \right ) {a}^{2}b \right ) } \]

____________________________________________________________________________________

73.4 Problem 4

problem number 672

Added Feb. 9, 2019.

Problem Chapter 3.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b) w_x +(c y +d) w_y = k x^3+n y^3 \]

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 + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == k*x^3 + n*y^3; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {6 a^4 c^4 c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )+2 a^4 c^3 n y^3-3 a^4 c^2 d n y^2+6 a^4 c d^2 n y+11 a^4 d^3 n-6 a^3 c d^3 n \log (a x+b)+2 a^3 c^4 k x^3-3 a^2 b c^4 k x^2-6 b^3 c^4 k \log (a x+b)+6 a b^2 c^4 k x}{6 a^4 c^4}\right \}\right \} \]

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+b)*diff(w(x,y),x) + (c*y+d)*diff(w(x,y),y) =k*x^3+n*y^3; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =1/3\,{\frac {k{x}^{3}}{a}}-1/2\,{\frac {k{x}^{2}b}{{a}^{2}}}+{\frac {{b}^{2}kx}{{a}^{3}}}+1/3\,{\frac {n{y}^{3}}{c}}-1/2\,{\frac {dn{y}^{2}}{{c}^{2}}}+{\frac {{d}^{2}ny}{{c}^{3}}}+1/6\,{\frac {1}{{a}^{4}{c}^{4}} \left ( 6\,{\it \_F1} \left ( {\frac {cy+d}{c} \left ( ax+b \right ) ^{-{\frac {c}{a}}}} \right ) {a}^{4}{c}^{4}-6\,\ln \left ( ax+b \right ) {d}^{3}n{a}^{3}c-6\,\ln \left ( ax+b \right ) {b}^{3}k{c}^{4}+11\,{a}^{4}{d}^{3}n \right ) } \]

____________________________________________________________________________________

73.5 Problem 5

problem number 673

Added Feb. 9, 2019.

Problem Chapter 3.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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] + x*y*D[w[x, y], y] == y^2*(a*x + b*y); 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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) + x*y*diff(w(x,y),y) =y^2*(a*x + b*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =1/2\,{y}^{2}a+{\it \_F1} \left ( {\frac {y}{x}} \right ) +1/2\,{\frac {b{y}^{3}}{x}} \]

____________________________________________________________________________________

73.6 Problem 6

problem number 674

Added Feb. 9, 2019.

Problem Chapter 3.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^3 w_x +b y^3 w_y = c x + d \]

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*x^3*D[w[x, y], x] + b*y^3*D[w[x, y], y] == c*x + d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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*x^3*diff(w(x,y),x) + b*y^3*diff(w(x,y),y) =c*x+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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