148 HFOPDE, chapter 5.8.2

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148.1 Problem 1

problem number 1164

Added April 13, 2019.

Problem Chapter 5.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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148.2 Problem 2

problem number 1165

Added April 13, 2019.

Problem Chapter 5.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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148.3 Problem 3

problem number 1166

Added April 13, 2019.

Problem Chapter 5.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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)*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;
 

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

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148.4 Problem 4

problem number 1167

Added April 13, 2019.

Problem Chapter 5.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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148.5 Problem 5

problem number 1168

Added April 13, 2019.

Problem Chapter 5.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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) = c*w(x,y)+f(x)*g(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;
 

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

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148.6 Problem 6

problem number 1169

Added April 13, 2019.

Problem Chapter 5.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) w_x + f_2(y) w_y = a w + g_1(x)+g_2(y) \]

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]*D[w[x, y], x] + f2[y]*D[w[x, y], y] == a*w[x,y]+g1[x]+g2[y]; 
 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 :=  f1(x)*diff(w(x,y),x)+ f2(y)*diff(w(x,y),y) = a*w(x,y)+g1(x)+g2(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;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\rm e}^{-a\int \! \left ( {\it f1} \left ( {\it \_f} \right ) \right ) ^{-1}\,{\rm d}{\it \_f}}} \left ( {\it g1} \left ( {\it \_f} \right ) +{\it g2} \left ( \RootOf \left ( \int \! \left ( {\it f1} \left ( {\it \_f} \right ) \right ) ^{-1}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\! \left ( {\it f2} \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-\int \! \left ( {\it f1} \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( {\it f2} \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \right ) \right ) }{{\it f1} \left ( {\it \_f} \right ) }}{d{\it \_f}}+{\it \_F1} \left ( -\int \! \left ( {\it f1} \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( {\it f2} \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \right ) {{\rm e}^{\int \!{\frac {a}{{\it f1} \left ( x \right ) }}\,{\rm d}x}} \]