94 HFOPDE, chapter 3.8.1

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94.1 Problem 1

problem number 791

Added Feb. 11, 2019.

Problem Chapter 3.8.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.2 Problem 2

problem number 792

Added Feb. 11, 2019.

Problem Chapter 3.8.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.3 Problem 3

problem number 793

Added Feb. 11, 2019.

Problem Chapter 3.8.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.4 Problem 4

problem number 794

Added Feb. 11, 2019.

Problem Chapter 3.8.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.5 Problem 5

problem number 795

Added Feb. 11, 2019.

Problem Chapter 3.8.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.6 Problem 6

problem number 796

Added Feb. 11, 2019.

Problem Chapter 3.8.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.7 Problem 7

problem number 797

Added Feb. 11, 2019.

Problem Chapter 3.8.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.8 Problem 8

problem number 798

Added Feb. 11, 2019.

Problem Chapter 3.8.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.9 Problem 9

problem number 799

Added Feb. 11, 2019.

Problem Chapter 3.8.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

\[ w \left ( x,y \right ) = \left ( -ab+c \right ) x+ab+by+{\it \_F1} \left ( \left ( y+a \right ) {{\rm e}^{-x}} \right ) \]

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94.10 Problem 10

problem number 800

Added Feb. 11, 2019.

Problem Chapter 3.8.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.11 Problem 11

problem number 801

Added Feb. 11, 2019.

Problem Chapter 3.8.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]; 
 pde = f[x]*D[w[x, y], x] + (y*g1[x] + g0[x])*D[w[x, y], y] == y^2*h2[x] + y*h1[x] + h0[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol[[2]] = Simplify[sol[[2]]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (y 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]\right )+\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,x\},\text {Assumptions}\to \text {True}\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,K[3]\},\text {Assumptions}\to \text {True}\right ]+y \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\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,x\},\text {Assumptions}\to \text {True}\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,K[3]\},\text {Assumptions}\to \text {True}\right ]+y \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right )\right )^2+\text {h0}(K[3])}{f(K[3])} \, dK[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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2'; 
pde := diff(w(x,y),x) + (y*g1(x)+g0(x))*diff(w(x,y),y) =   y^2*h2(x)+y*h1(x)+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) =\int ^{x}\!{\it h2} \left ( {\it \_f} \right ) {{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( \int \!{\it g0} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \right ) ^{2}+2\,{\it h2} \left ( {\it \_f} \right ) {{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\int \!{\it g0} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f} \left ( -\int \!{\it g0} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) +{\it h2} \left ( {\it \_f} \right ) {{\rm e}^{2\,\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( -\int \!{\it g0} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) ^{2}+{{\rm e}^{\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h1} \left ( {\it \_f} \right ) \int \!{\it g0} \left ( {\it \_f} \right ) {{\rm e}^{-\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}\,{\rm d}{\it \_f}+{{\rm e}^{\int \!{\it g1} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\it h1} \left ( {\it \_f} \right ) \left ( -\int \!{\it g0} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) +{\it h0} \left ( {\it \_f} \right ) {d{\it \_f}}+{\it \_F1} \left ( -\int \!{\it g0} \left ( x \right ) {{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+y{{\rm e}^{-\int \!{\it g1} \left ( x \right ) \,{\rm d}x}} \right ) \]

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94.12 Problem 12

problem number 802

Added Feb. 11, 2019.

Problem Chapter 3.8.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]; 
 pde = f[x]*D[w[x, y], x] + (y*g1[x] + y^k*g2[x])*D[w[x, y], y] == h[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol[[2]] = Simplify[sol[[2]]];
 

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

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

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

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94.13 Problem 13

problem number 803

Added Feb. 11, 2019.

Problem Chapter 3.8.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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94.14 Problem 14

problem number 804

Added Feb. 11, 2019.

Problem Chapter 3.8.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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94.15 Problem 15

problem number 805

Added Feb. 11, 2019.

Problem Chapter 3.8.1.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ y^k f(x) w_x + (y^{k+1} g_1(x) + g_0(x)) w_y = y^{3 k +1} h_2(x) + y^{2 k+1} h_1(x) + y^k h_0(x) \]

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]; 
 pde = y^k*f[x]*D[w[x, y], x] + (y^(k + 1)*g1[x] + g0[x])*D[w[x, y], y] == y^(3*k + 1)*h2[x] + y^(2*k + 1)*h1[x] + y^k*h0[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol[[2]] = Simplify[sol[[2]]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (y^{k+1} \exp \left (-(k+1) \int _1^x \frac {\text {g1}(K[1])}{f(K[1])} \, dK[1]\right )-(k+1) \int _1^x \frac {\text {g0}(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]\right )+\int _1^x \frac {\text {h1}(K[3]) \left (\left (\exp \left (-(k+1) \left (\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text {Assumptions}\to \text {True}\right ]\right )\right ) \left (-(k+1) \exp \left ((k+1) \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 (-(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])},\{K[2],1,x\},\text {Assumptions}\to \text {True}\right ]+(k+1) \text {Integrate}\left [\frac {\text {g0}(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])},\{K[2],1,K[3]\},\text {Assumptions}\to \text {True}\right ] \exp \left ((k+1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right )+y^{k+1}\right )\right )^{\frac {1}{k+1}}\right )^{k+1}+\text {h2}(K[3]) \left (\left (\exp \left (-(k+1) \left (\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[3]\},\text {Assumptions}\to \text {True}\right ]\right )\right ) \left (-(k+1) \exp \left ((k+1) \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 (-(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])},\{K[2],1,x\},\text {Assumptions}\to \text {True}\right ]+(k+1) \text {Integrate}\left [\frac {\text {g0}(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])},\{K[2],1,K[3]\},\text {Assumptions}\to \text {True}\right ] \exp \left ((k+1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right )+y^{k+1}\right )\right )^{\frac {1}{k+1}}\right )^{2 k+1}+\text {h0}(K[3])}{f(K[3])} \, dK[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';q:='q';p:='p';l:='l'; 
g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2'; 
pde := y^k*f(x)*diff(w(x,y),x) +(y^(k+1)* g1(x) + g0(x))*diff(w(x,y),y) =   y^(3*k +1)*h2(x) + y^(2*k+1)*h1(x) + y^k*h0(x); 
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}{f \left ( {\it \_f} \right ) } \left ( {\it h0} \left ( {\it \_f} \right ) + \left ( \left ( k\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\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 \left ( k+1 \right ) }}}\,{\rm d}x-\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 \left ( k+1 \right ) }}}\,{\rm d}x+\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) ^{k} \left ( k\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\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 \left ( k+1 \right ) }}}\,{\rm d}x-\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 \left ( k+1 \right ) }}}\,{\rm d}x+\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\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 ( \left ( k\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\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 \left ( k+1 \right ) }}}\,{\rm d}x-\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 \left ( k+1 \right ) }}}\,{\rm d}x+\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) ^{2\,k} \left ( k\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f}+{y}^{k+1}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\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 \left ( k+1 \right ) }}}\,{\rm d}x-\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 \left ( k+1 \right ) }}}\,{\rm d}x+\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} \left ( k+1 \right ) }}}\,{\rm d}{\it \_f} \right ) ^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}{\it h2} \left ( {\it \_f} \right ) \right ) }{d{\it \_f}}+{\it \_F1} \left ( {y}^{k+1}{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x \left ( k+1 \right ) }}-k\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 \left ( k+1 \right ) }}}\,{\rm d}x-\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 \left ( k+1 \right ) }}}\,{\rm d}x \right ) \]

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94.16 Problem 16

problem number 806

Added Feb. 11, 2019.

Problem Chapter 3.8.1.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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