96 HFOPDE, chapter 3.8.3

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96.1 Problem 1

problem number 814

Added Feb. 11, 2019.

Problem Chapter 3.8.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = f(\alpha x+\beta y) \]

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, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == f[alpha*x + beta*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \int _1^x \frac {f\left (\frac {\beta (b K[1]+a y-b x)}{a}+\alpha K[1]\right )}{a} \, dK[1]+c_1\left (\frac {a y-b x}{a}\right )\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*diff(w(x,y),x) +b*diff(w(x,y),y) =  f(alpha*x+beta*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a}f \left ( {\frac { \left ( ya-bx \right ) \beta +{\it \_a}\,a\alpha +{\it \_a}\,b\beta }{a}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

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96.2 Problem 2

problem number 815

Added Feb. 11, 2019.

Problem Chapter 3.8.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = x f(\frac {y}{x}) \]

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, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = x*D[w[x, y], x] + y*D[w[x, y], y] == x*f[y/x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right )+x f\left (\frac {y}{x}\right )\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  x*diff(w(x,y),x) +y*diff(w(x,y),y) =  x*f(y/x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =xf \left ( {\frac {y}{x}} \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) \]

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96.3 Problem 3

problem number 816

Added Feb. 11, 2019.

Problem Chapter 3.8.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = x*D[w[x, y], x] + y*D[w[x, y], y] == f[x^2 + y^2]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \int _1^x \frac {f\left (\frac {y^2 K[1]^2}{x^2}+K[1]^2\right )}{K[1]} \, dK[1]+c_1\left (\frac {y}{x}\right )\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  x*diff(w(x,y),x) +y*diff(w(x,y),y) =  f(x^2+y^2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{{\it \_a}}f \left ( {\frac {{y}^{2}{{\it \_a}}^{2}}{{x}^{2}}}+{{\it \_a}}^{2} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]

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96.4 Problem 4

problem number 817

Added Feb. 11, 2019.

Problem Chapter 3.8.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = x f(\frac {y}{x})+ g(x^2+y^2) \]

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, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = x*D[w[x, y], x] + y*D[w[x, y], y] == x*f[y/x] + g[x^2 + y^2]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \int _1^x \frac {K[1] f\left (\frac {y}{x}\right )+g\left (\frac {y^2 K[1]^2}{x^2}+K[1]^2\right )}{K[1]} \, dK[1]+c_1\left (\frac {y}{x}\right )\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  x*diff(w(x,y),x) +y*diff(w(x,y),y) =  x*f(y/x)+g(x^2+y^2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{{\it \_a}} \left ( {\it \_a}\,f \left ( {\frac {y}{x}} \right ) +g \left ( {\frac {{y}^{2}{{\it \_a}}^{2}}{{x}^{2}}}+{{\it \_a}}^{2} \right ) \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]

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96.5 Problem 5

problem number 818

Added Feb. 11, 2019.

Problem Chapter 3.8.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == x^k*f[x^n*x^m]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \int _1^x \frac {K[1]^{k-1} f\left (K[1]^{m+n}\right )}{a} \, dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  a*x*diff(w(x,y),x) +b*y*diff(w(x,y),y) =  x^k*f(x^n*y^m); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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96.6 Problem 6

problem number 819

Added Feb. 11, 2019.

Problem Chapter 3.8.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = m*x*D[w[x, y], x] + n*y*D[w[x, y], y] == f[a*x^n + b*x^m]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \int _1^x \frac {f\left (a K[1]^n+b K[1]^m\right )}{m K[1]} \, dK[1]+c_1\left (y x^{-\frac {n}{m}}\right )\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  m*x*diff(w(x,y),x) +n*y*diff(w(x,y),y) =  f(a*x^n+b*y^m); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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96.7 Problem 7

problem number 820

Added Feb. 17, 2019.

Problem Chapter 3.8.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + x y w_y = y^k f(\alpha x + \beta y) \]

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, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = x^2*D[w[x, y], x] + x*y*D[w[x, y], y] == y^k*f[alpha*x + beta*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \int _1^x \frac {\left (\frac {y K[1]}{x}\right )^k f\left (\alpha K[1]+\frac {\beta y K[1]}{x}\right )}{K[1]^2} \, dK[1]+c_1\left (\frac {y}{x}\right )\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde :=  x^2*diff(w(x,y),x) +x*y*diff(w(x,y),y) =  y^k*f(alpha*x+beta*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{{{\it \_a}}^{2}}f \left ( {\it \_a}\, \left ( {\frac {\beta \,y}{x}}+\alpha \right ) \right ) \left ( {\frac {y{\it \_a}}{x}} \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]

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96.8 Problem 8

problem number 821

Added Feb. 17, 2019.

Problem Chapter 3.8.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \frac {f(x)}{f'(x)} w_x + \frac {g(y)}{g'(y)} w_y = h(f(x)+g(y)) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 pde = (f[x]*D[w[x, y], x])/Derivative[1][f][x] + (g[y]*D[w[x, y], y])/Derivative[1][g][y] == h[f[x] + g[y]]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\log \left (\frac {\text {InverseFunction}\left [g^{(-1)},1,1\right ][y]}{f(x)}\right )\right )+\int _1^x \frac {f'(K[1]) h\left (g\left (\text {InverseFunction}\left [\text {InverseFunction}\left [g^{(-1)},1,1\right ],1,1\right ]\left [\frac {f(K[1]) \text {InverseFunction}\left [g^{(-1)},1,1\right ][y]}{f(x)}\right ]\right )+f(K[1])\right )}{f(K[1])} \, dK[1]\right \}\right \} \]

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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 
pde := f(x)/diff(f(x),x)*diff(w(x,y),x) +g(y)/diff(g(y),y)*diff(w(x,y),y) =  h(f(x)+g(y)); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {{\frac {\rm d}{{\rm d}{\it \_a}}}f \left ( {\it \_a} \right ) }{f \left ( {\it \_a} \right ) }h \left ( f \left ( {\it \_a} \right ) \left ( {\frac {g \left ( y \right ) }{f \left ( x \right ) }}+1 \right ) \right ) }{d{\it \_a}}+{\it \_F1} \left ( \ln \left ( {\frac {g \left ( y \right ) }{f \left ( x \right ) }} \right ) \right ) \]