147 HFOPDE, chapter 5.8.1

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147.1 Problem 1

problem number 1152

Added April 13, 2019.

Problem Chapter 5.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == f[x]*w[x,y]+g[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = f(x)*w(x,y)+g(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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147.2 Problem 2

problem number 1153

Added April 13, 2019.

Problem Chapter 5.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*y+k)*w[x,y]+f[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*y+k)*w(x,y)+f(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {f \left ( {\it \_a} \right ) }{a}{{\rm e}^{-{\frac { \left ( \left ( cy+k \right ) a-cb \left ( x-{\it \_a}/2 \right ) \right ) {\it \_a}}{{a}^{2}}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac { \left ( \left ( cy+k \right ) a-1/2\,bcx \right ) x}{{a}^{2}}}}} \]

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147.3 Problem 3

problem number 1154

Added April 13, 2019.

Problem Chapter 5.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == f[x]*y*w[x,y]+g[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = f(x)*y*w(x,y)+g(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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147.4 Problem 4

problem number 1155

Added April 13, 2019.

Problem Chapter 5.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == f[x]*w[x,y]+g[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = f(x)*w(x,y)+g(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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147.5 Problem 5

problem number 1156

Added April 13, 2019.

Problem Chapter 5.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f[x]*D[w[x, y], x] + (a+y+b)*D[w[x, y], y] == c*w[x,y]+g[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f(x)*diff(w(x,y),x)+ (a*y+b)*diff(w(x,y),y) = c*w(x,y)+g(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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147.6 Problem 6

problem number 1157

Added April 13, 2019.

Problem Chapter 5.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + g(x) w_y = h(x) w + p(x) \]

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f[x]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x,y]+p[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f(x)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) = h(x)*w(x,y)+p(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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147.7 Problem 7

problem number 1158

Added April 13, 2019.

Problem Chapter 5.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x])*D[w[x, y], y] == h1[x]*w[x,y]+h0[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\int _1^x \frac {\text {h1}(K[3])}{f(K[3])} \, dK[3]} \left (c_1\left (y e^{-\int _1^x \frac {\text {g1}(K[1])}{f(K[1])} \, dK[1]}-\int _1^x \frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])} \, dK[2]\right )+\int _1^x \frac {\text {h0}(K[4]) \exp \left (-\text {Integrate}\left [\frac {\text {h1}(K[3])}{f(K[3])},\{K[3],1,K[4]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[4])} \, dK[4]\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) = h1(x)*w(x,y)+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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147.8 Problem 8

problem number 1159

Added April 13, 2019.

Problem Chapter 5.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x) y+ g_0(x)) w_y = h_2(x) w + h_1(x) y + h0(x) \]

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x])*D[w[x, y], y] == h2[x]*w[x,y]+h1[x]*y+h0[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\int _1^x \frac {\text {h2}(K[3])}{f(K[3])} \, dK[3]} \left (c_1\left (y e^{-\int _1^x \frac {\text {g1}(K[1])}{f(K[1])} \, dK[1]}-\int _1^x \frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])} \, dK[2]\right )+\int _1^x \frac {\exp \left (-\text {Integrate}\left [\frac {\text {h2}(K[3])}{f(K[3])},\{K[3],1,K[4]\},\text {Assumptions}\to \text {True}\right ]\right ) \left (\text {h1}(K[4]) \exp \left (\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[4]\},\text {Assumptions}\to \text {True}\right ]\right ) \left (-\text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text {Assumptions}\to \text {True}\right ]+\text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,K[4]\},\text {Assumptions}\to \text {True}\right ]+y \exp \left (-\text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right )\right )+\text {h0}(K[4])\right )}{f(K[4])} \, dK[4]\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) = h2(x)*w(x,y)+h1(x)*y+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{f \left ( {\it \_g} \right ) } \left ( {\it h1} \left ( {\it \_g} \right ) \left ( y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}-\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+\int \!{\frac {{\it g0} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}}\,{\rm d}{\it \_g} \right ) {{\rm e}^{-\int \!{\frac {{\it h2} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}+\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}+{{\rm e}^{-\int \!{\frac {{\it h2} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}{\it h0} \left ( {\it \_g} \right ) \right ) }{d{\it \_g}}+{\it \_F1} \left ( -\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+y{{\rm e}^{-\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) \right ) {{\rm e}^{\int \!{\frac {{\it h2} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \]

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147.9 Problem 9

problem number 1160

Added April 13, 2019.

Problem Chapter 5.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x) y+ g_0(x) y^k) w_y = h_2(x) w + h_1(x) y^n + h0(x) \]

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*y^k)*D[w[x, y], y] == h2[x]*w[x,y]+h1[x]*y^n+h0[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\int _1^x \frac {\text {h2}(K[3])}{f(K[3])} \, dK[3]} \left (c_1\left ((k-1) \int _1^x \frac {\text {g0}(K[2]) \exp \left ((k-1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])} \, dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x \frac {\text {g1}(K[1])}{f(K[1])} \, dK[1]\right )\right )+\int _1^x \frac {\exp \left (-\text {Integrate}\left [\frac {\text {h2}(K[3])}{f(K[3])},\{K[3],1,K[4]\},\text {Assumptions}\to \text {True}\right ]\right ) \left (\text {h1}(K[4]) \left (\left (y^{-k} \exp \left (-(k-1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[4]\},\text {Assumptions}\to \text {True}\right ]\right ) \left ((k-1) y^k \text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left ((k-1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,x\},\text {Assumptions}\to \text {True}\right ]-(k-1) y^k \text {Integrate}\left [\frac {\text {g0}(K[2]) \exp \left ((k-1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]\right )}{f(K[2])},\{K[2],1,K[4]\},\text {Assumptions}\to \text {True}\right ]+y \exp \left ((k-1) \text {Integrate}\left [\frac {\text {g1}(K[1])}{f(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]\right )\right )\right )^{\frac {1}{1-k}}\right )^n+\text {h0}(K[4])\right )}{f(K[4])} \, dK[4]\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x)*y^k)*diff(w(x,y),y) = h2(x)*w(x,y)+h1(x)*y^n+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{f \left ( {\it \_g} \right ) }{{\rm e}^{-\int \!{\frac {{\it h2} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \left ( {\it h1} \left ( {\it \_g} \right ) \left ( \left ( \left ( 1-k \right ) \int \!{\frac {{\it g0} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}}\,{\rm d}{\it \_g}+ \left ( k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \right ) ^{n}+{\it h0} \left ( {\it \_g} \right ) \right ) }{d{\it \_g}}+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{1-k}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \right ) \right ) {{\rm e}^{\int \!{\frac {{\it h2} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \]

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147.10 Problem 10

problem number 1161

Added April 13, 2019.

Problem Chapter 5.8.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + (g_1(x)+ g_0(x) e^{\lambda y}) w_y = h_2(x) w + h_1(x) e^{\beta y} + h0(x) \]

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f[x]*D[w[x, y], x] + (g1[x]+g0[x]*Exp[lambda*y])*D[w[x, y], y] == h2[x]*w[x,y]+h1[x]*Exp[beta*y]+h0[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {Failed} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)+g0(x)*exp(lambda*y))*diff(w(x,y),y) = h2(x)*w(x,y)+h1(x)*exp(beta*y)+h0(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{f \left ( {\it \_g} \right ) } \left ( {\it h1} \left ( {\it \_g} \right ) \left ( \left ( \lambda \,\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{\lambda \,\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x-\int \!{\frac {{\it g0} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }{{\rm e}^{\lambda \,\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}}\,{\rm d}{\it \_g}\lambda +{{\rm e}^{\lambda \, \left ( \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x-y \right ) }} \right ) ^{-1} \right ) ^{{\frac {\beta }{\lambda }}}{{\rm e}^{-\int \!{\frac {{\it h2} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}+\beta \,\int \!{\frac {{\it g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}+{{\rm e}^{-\int \!{\frac {{\it h2} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}}{\it h0} \left ( {\it \_g} \right ) \right ) }{d{\it \_g}}+{\it \_F1} \left ( {\frac {1}{\lambda } \left ( -\lambda \,\int \!{\frac {{\it g0} \left ( x \right ) }{f \left ( x \right ) }{{\rm e}^{\lambda \,\int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x-{{\rm e}^{\lambda \, \left ( \int \!{\frac {{\it g1} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x-y \right ) }} \right ) } \right ) \right ) {{\rm e}^{\int \!{\frac {{\it h2} \left ( x \right ) }{f \left ( x \right ) }}\,{\rm d}x}} \]

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147.11 Problem 11

problem number 1162

Added April 13, 2019.

Problem Chapter 5.8.1.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) y^k w_x + f_2(x) w_y = g(x) w + h(x) \]

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f1[x]*y^k*D[w[x, y], x] + f2[x]*D[w[x, y], y] == g[x]*w[x,y]+h[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (\int _1^x \frac {g(K[2]) \left (\left (-(k+1) \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]+(k+1) \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]+y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k}}{\text {f1}(K[2])} \, dK[2]\right ) \left (\int _1^x \frac {h(K[3]) \left (\left (-(k+1) \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]+(k+1) \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,K[3]\},\text {Assumptions}\to \text {True}\right ]+y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k} \exp \left (-\text {Integrate}\left [\frac {g(K[2]) \left (\left (-(k+1) \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]+(k+1) \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]+y^{k+1}\right )^{\frac {1}{k+1}}\right )^{-k}}{\text {f1}(K[2])},\{K[2],1,K[3]\},\text {Assumptions}\to \text {True}\right ]\right )}{\text {f1}(K[3])} \, dK[3]+c_1\left (\frac {y^{k+1}}{k+1}-\int _1^x \frac {\text {f2}(K[1])}{\text {f1}(K[1])} \, dK[1]\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f1(x)*y^k*diff(w(x,y),x)+ f2(x)*diff(w(x,y),y) = g(x)*w(x,y)+h(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {h \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) } \left ( \left ( \left ( k+1 \right ) \int \!{\frac {{\it f2} \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}+ \left ( -1-k \right ) \int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+{y}^{k}y \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}{{\rm e}^{-\int \!{\frac {g \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) } \left ( \left ( \left ( k+1 \right ) \int \!{\frac {{\it f2} \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}+ \left ( -1-k \right ) \int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+{y}^{k}y \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}\,{\rm d}{\it \_f}}}}{d{\it \_f}}+{\it \_F1} \left ( \left ( -1-k \right ) \int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+{y}^{k}y \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {g \left ( {\it \_b} \right ) }{{\it f1} \left ( {\it \_b} \right ) } \left ( \left ( \left ( k+1 \right ) \int \!{\frac {{\it f2} \left ( {\it \_b} \right ) }{{\it f1} \left ( {\it \_b} \right ) }}\,{\rm d}{\it \_b}+ \left ( -1-k \right ) \int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+{y}^{k}y \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_b}}}} \]

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147.12 Problem 12

problem number 1163

Added April 13, 2019.

Problem Chapter 5.8.1.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) e^{\lambda y} w_x + f_2(x) w_y = g(x) w + h(x) \]

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,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = f1[x]*Exp[lambda*y]*D[w[x, y], x] + f2[x]*D[w[x, y], y] == g[x]*w[x,y]+h[x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (\int _1^x \frac {g(K[2])}{\text {f1}(K[2]) \left (-\lambda \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]+\lambda \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]+e^{\lambda y}\right )} \, dK[2]\right ) \left (\int _1^x \frac {h(K[3]) \exp \left (-\text {Integrate}\left [\frac {g(K[2])}{\text {f1}(K[2]) \left (-\lambda \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]+\lambda \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,K[2]\},\text {Assumptions}\to \text {True}\right ]+e^{\lambda y}\right )},\{K[2],1,K[3]\},\text {Assumptions}\to \text {True}\right ]\right )}{\text {f1}(K[3]) \left (-\lambda \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,x\},\text {Assumptions}\to \text {True}\right ]+\lambda \text {Integrate}\left [\frac {\text {f2}(K[1])}{\text {f1}(K[1])},\{K[1],1,K[3]\},\text {Assumptions}\to \text {True}\right ]+e^{\lambda y}\right )} \, dK[3]+c_1\left (\frac {e^{\lambda y}}{\lambda }-\int _1^x \frac {\text {f2}(K[1])}{\text {f1}(K[1])} \, dK[1]\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  f1(x)*exp(lambda*y)*diff(w(x,y),x)+ f2(x)*diff(w(x,y),y) = g(x)*w(x,y)+h(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {h \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) }{{\rm e}^{-{\frac {1}{\lambda }\int \!{\frac {g \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) } \left ( \int \!{\frac {{\it f2} \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}+{\frac {1}{\lambda } \left ( {{\rm e}^{y\lambda }}-\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x\lambda \right ) } \right ) ^{-1}}\,{\rm d}{\it \_f}}}} \left ( -\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x\lambda +\int \!{\frac {{\it f2} \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}\lambda +{{\rm e}^{y\lambda }} \right ) ^{-1}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{\lambda } \left ( {{\rm e}^{y\lambda }}-\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x\lambda \right ) } \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {g \left ( {\it \_b} \right ) }{{\it f1} \left ( {\it \_b} \right ) } \left ( -\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x\lambda +\int \!{\frac {{\it f2} \left ( {\it \_b} \right ) }{{\it f1} \left ( {\it \_b} \right ) }}\,{\rm d}{\it \_b}\lambda +{{\rm e}^{y\lambda }} \right ) ^{-1}}{d{\it \_b}}}} \]