121 HFOPDE, chapter 4.8.1

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121.1 Problem 1

problem number 969

Added March 10, 2019.

Problem Chapter 4.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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121.2 Problem 2

problem number 970

Added March 10, 2019.

Problem Chapter 4.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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121.3 Problem 3

problem number 971

Added March 10, 2019.

Problem Chapter 4.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ) {{\rm e}^{\int ^{x}\!f \left ( {\it \_a} \right ) {a}^{2}{{\it \_a}}^{2}+2\,f \left ( {\it \_a} \right ) \left ( -ax+y \right ) a{\it \_a}+g \left ( {\it \_a} \right ) a{\it \_a}+f \left ( {\it \_a} \right ) \left ( -ax+y \right ) ^{2}+g \left ( {\it \_a} \right ) \left ( -ax+y \right ) +h \left ( {\it \_a} \right ) {d{\it \_a}}}} \]

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121.4 Problem 4

problem number 972

Added March 10, 2019.

Problem Chapter 4.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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121.5 Problem 5

problem number 973

Added March 10, 2019.

Problem Chapter 4.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = f(x) e^{\lambda y} w \]

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 = D[w[x, y], x] + a*D[w[x, y], y] == f[x]*Exp[lambda*y]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -ax+y \right ) {{\rm e}^{\int ^{x}\!f \left ( {\it \_a} \right ) {{\rm e}^{{\it \_a}\,a\lambda + \left ( -ax+y \right ) \lambda }}{d{\it \_a}}}} \]

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121.6 Problem 6

problem number 974

Added March 10, 2019.

Problem Chapter 4.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + f(x)) w_y = g(x) w \]

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

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

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

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

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121.7 Problem 7

problem number 975

Added March 10, 2019.

Problem Chapter 4.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y + f(x)) w_y = g(x) y^k w \]

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

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

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

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

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121.8 Problem 8

problem number 976

Added March 10, 2019.

Problem Chapter 4.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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121.9 Problem 9

problem number 977

Added March 10, 2019.

Problem Chapter 4.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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121.10 Problem 10

problem number 978

Added March 10, 2019.

Problem Chapter 4.8.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (y+a x) w_y = g(x) w \]

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 = f[x]*D[w[x, y], x] + (y + a*x)*D[w[x, y], y] == g[x]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\int _1^x \frac {g(K[3])}{f(K[3])} \, dK[3]} c_1\left (y e^{-\int _1^x \frac {1}{f(K[1])} \, dK[1]}-\int _1^x \frac {a K[2] \exp \left (-\text {Integrate}\left [\frac {1}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])} \, dK[2]\right )\right \}\right \} \]

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

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

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121.11 Problem 11

problem number 979

Added March 10, 2019.

Problem Chapter 4.8.1.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x) y+g_0(x)) w_y = \left ( h_2(x) y^2+ h_1(x)y + h_0(x) \right ) w \]

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] + (g1[x]*y + g0[x])*D[w[x, y], y] == (h2[x]*y^2 + h1[x]*y + h0[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 (-e^{-\int _1^x \frac {\text {g1}(K[1])}{f(K[1])} \, dK[1]} \left (e^{\int _1^x \frac {\text {g1}(K[1])}{f(K[1])} \, dK[1]} \int _1^x \frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])} \, dK[2]-y\right )\right ) \exp \left (\int _1^x \frac {\text {h1}(K[3]) \exp \left (\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text {Assumptions}\to \text {True}\right ]\right ) \left (\text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,K[3]\},\text {Assumptions}\to \text {True}\right ]-\exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right ) \left (\exp \left (\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right ) \text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text {Assumptions}\to \text {True}\right ]-y\right )\right )+\text {h2}(K[3]) \exp \left (2 \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text {Assumptions}\to \text {True}\right ]\right ) \left (\text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,K[3]\},\text {Assumptions}\to \text {True}\right ]-\exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right ) \left (\exp \left (\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right ) \text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text {Assumptions}\to \text {True}\right ]-y\right )\right )^2+\text {h0}(K[3])}{f(K[3])} \, dK[3]\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(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) =(h2(x)*y^2+h1(x)*y+h0(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 ( -\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{f \left ( {\it \_f} \right ) } \left ( \left ( \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}}\,{\rm d}{\it \_f} \right ) ^{2}{{\rm e}^{2\,\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h2} \left ( {\it \_f} \right ) +2\,\int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}}\,{\rm d}{\it \_f}{{\rm e}^{2\,\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h2} \left ( {\it \_f} \right ) \left ( -\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) +{{\rm e}^{2\,\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h2} \left ( {\it \_f} \right ) \left ( -\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) ^{2}+{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h1} \left ( {\it \_f} \right ) \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}}\,{\rm d}{\it \_f}+{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h1} \left ( {\it \_f} \right ) \left ( -\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) +{\it h0} \left ( {\it \_f} \right ) \right ) }{d{\it \_f}}}} \]

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121.12 Problem 12

problem number 980

Added March 10, 2019.

Problem Chapter 4.8.1.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x) y+g_2(x) y^k) w_y = h(x) w \]

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 = f[x]*D[w[x, y], x] + (g1[x]*y + g2[x]*y^k)*D[w[x, y], y] == h[x]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\int _1^x \frac {h(K[3])}{f(K[3])} \, dK[3]} c_1\left ((k-1) \int _1^x \frac {\text {g2}(K[2]) \exp \left ((k-1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])} \, dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x \frac {\text {g1}(K[1])}{f(K[1])} \, dK[1]\right )\right )\right \}\right \} \]

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(w(x,y),x)+ (g1(x)*y+g2(x)*y^k)*diff(w(x,y),y) =h(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol := simplify(sol);
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \left ( k-1 \right ) \int \!{\frac {{\it g2} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) {{\rm e}^{\int \!{\frac {h \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \]

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121.13 Problem 13

problem number 981

Added March 10, 2019.

Problem Chapter 4.8.1.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x)+g_2(x) e^{\lambda y}) w_y = h(x) w \]

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 = f[x]*D[w[x, y], x] + (g1[x]*y + g2[x]*Exp[lambda*y])*D[w[x, y], y] == h[x]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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(w(x,y),x)+ (g1(x)*y+g2(x)*exp(lambda*y))*diff(w(x,y),y) =h(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

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121.14 Problem 14

problem number 982

Added March 10, 2019.

Problem Chapter 4.8.1.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) y^k w_x + g(x) w_y = h(x) w \]

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 = f[x]*y^k*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

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

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121.15 Problem 15

problem number 983

Added March 10, 2019.

Problem Chapter 4.8.1.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) e^{\lambda y} w_x + g(x) w_y = h(x) w \]

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 = f[x]*Exp[lambda*y]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x, y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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)*exp(lambda*y)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) =h(x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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