123 HFOPDE, chapter 4.8.3

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123.1 Problem 1

problem number 991

Added March 10, 2019.

Problem Chapter 4.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

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';t:='t'; 
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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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123.2 Problem 2

problem number 992

Added March 10, 2019.

Problem Chapter 4.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

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';t:='t'; 
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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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123.3 Problem 3

problem number 993

Added March 10, 2019.

Problem Chapter 4.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

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';t:='t'; 
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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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123.4 Problem 4

problem number 994

Added March 10, 2019.

Problem Chapter 4.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

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';t:='t'; 
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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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123.5 Problem 5

problem number 995

Added March 10, 2019.

Problem Chapter 4.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

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';t:='t'; 
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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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123.6 Problem 6

problem number 996

Added March 10, 2019.

Problem Chapter 4.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

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';t:='t'; 
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)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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123.7 Problem 7

problem number 997

Added March 10, 2019.

Problem Chapter 4.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1\left (y-\int _1^x \frac {g(K[1]) f'(K[1])}{f(K[1]) g'(K[1])} \, dK[1]\right ) \exp \left (\int _1^x \frac {f'(K[2]) h\left (g\left (-\text {Integrate}\left [\frac {g(K[1]) f'(K[1])}{f(K[1]) g'(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]+\text {Integrate}\left [\frac {g(K[1]) f'(K[1])}{f(K[1]) g'(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]+y\right )+f(K[2])\right )}{f(K[2])} \, dK[2]\right )\right \}\right \} \]

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';t:='t'; 
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(x)/diff(g(x),x)*diff(w(x,y),y) = h(f(x)+g(y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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