HFOPDE, chapter 5.8.3

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149.1 Problem 1

Problem Chapter 5.8.3.1, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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 :=  x*diff(w(x,y),x)+ y*diff(w(x,y),y) = x*f(y/x)*w(x,y)+g(x,y); 
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;

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

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149.2 Problem 2

Problem Chapter 5.8.3.2, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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,y)*w(x,y)+g(x,y); 
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;

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

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149.3 Problem 3

Problem Chapter 5.8.3.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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,F]; 
 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,y]*w[x,y]+F[x,y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];

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

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,y)*w(x,y)+F(x,y); 
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;

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

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149.4 Problem 4

Problem Chapter 5.8.3.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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,F]; 
 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] == h[x,y]*w[x,y]+F[x,y]; 
 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 {h\left (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 )\right )}{f(K[3])} \, dK[3]\right ) \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 {F\left (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 )\right ) \exp \left (-\text {Integrate}\left [\frac {h\left (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 )\right )}{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) = h(x,y)*w(x,y)+F(x,y); 
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 ) }F \left ( {\it \_g}, \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 g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \right ) {{\rm e}^{-\int \!{\frac {1}{f \left ( {\it \_g} \right ) }h \left ( {\it \_g}, \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 g1} \left ( {\it \_g} \right ) }{f \left ( {\it \_g} \right ) }}\,{\rm d}{\it \_g}}} \right ) }\,{\rm d}{\it \_g}}}}{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 ^{x}\!{\frac {1}{f \left ( {\it \_f} \right ) }h \left ( {\it \_f}, \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}-\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 \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) }{d{\it \_f}}}} \]

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149.5 Problem 5

Problem Chapter 5.8.3.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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,F]; 
 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] == h[x,y]*w[x,y]+F[x,y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];

\[ \text {\$Aborted} \]

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) = h(x,y)*w(x,y)+F(x,y); 
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 ) }F \left ( {\it \_g}, \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 ) {{\rm e}^{-\int \!{\frac {1}{f \left ( {\it \_g} \right ) }h \left ( {\it \_g}, \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 ) }\,{\rm d}{\it \_g}}}}{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 ^{x}\!{\frac {1}{f \left ( {\it \_f} \right ) }h \left ( {\it \_f}, \left ( \left ( 1-k \right ) \int \!{\frac {{\it g0} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it g1} \left ( {\it \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}}}\,{\rm d}{\it \_f}+ \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 \_f} \right ) }{f \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}}} \right ) }{d{\it \_f}}}} \]

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149.6 Problem 6

Problem Chapter 5.8.3.6, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

149 HFOPDE, chapter 5.8.3

149.1 Problem 1

149.2 Problem 2

149.3 Problem 3

149.4 Problem 4

149.5 Problem 5

149.6 Problem 6