HFOPDE, chapter 5.7.2

____________________________________________________________________________________

144.1 Problem 1

Problem Chapter 5.7.2.1, 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 = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcCos[lambda*x]^k+c2*ArcCos[beta*y]^n; 
 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 :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  w(x,y)+c1*arccos(lambda*x)^k+c2*arccos(beta*y)^n; 
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;

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c1}\, \left ( \arccos \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}+{\it c2}\, \left ( \arccos \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}} \]

____________________________________________________________________________________

144.2 Problem 2

Problem Chapter 5.7.2.2, 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*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcCos[lambda*x]^k*ArcCos[beta*y]^n; 
 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 :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =  c*w(x,y)+ arccos(lambda*x)^k*arccos(beta*y)^n; 
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;

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

____________________________________________________________________________________

144.3 Problem 3

Problem Chapter 5.7.2.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

\[ a w_x + b w_y = \left ( c_1 \arccos (\lambda _1 x) + c_2 \arccos (\lambda _2 y)\right ) w+ s_1 \arccos ^n(\beta _1 x)+ s_2 \arccos ^k(\beta _2 y) \]

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 = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcCos[lambda1*x] + c2*ArcCos[lambda2*y])*w[x,y]+ s1*ArcCos[beta1*x]^n+ s2*ArcCos[beta2*y]^k; 
 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 :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arccos(lambda1*x) + c2*arccos(lambda2*y))*w(x,y)+ s1*arccos(beta1*x)^n+ s2*arccos(beta2*y)^k; 
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;

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {1}{a\lambda 1\,\lambda 2\,b} \left ( \sqrt {-{\frac { \left ( \left ( \lambda 2\,y-1 \right ) a-b\lambda 2\, \left ( x-{\it \_a} \right ) \right ) \left ( \left ( \lambda 2\,y+1 \right ) a-b\lambda 2\, \left ( x-{\it \_a} \right ) \right ) }{{a}^{2}}}}a{\it c2}\,\lambda 1-\lambda 2\, \left ( \left ( \left ( {\it \_a}-x \right ) b+ya \right ) {\it c2}\,\lambda 1\,\arccos \left ( {\frac {\lambda 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) +b{\it c1}\, \left ( \arccos \left ( \lambda 1\,{\it \_a} \right ) {\it \_a}\,\lambda 1-\sqrt {-{{\it \_a}}^{2}{\lambda 1}^{2}+1} \right ) \right ) \right ) }}} \left ( {\it s2}\, \left ( \arccos \left ( {\frac {\beta 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{k}+{\it s1}\, \left ( \arccos \left ( \beta 1\,{\it \_a} \right ) \right ) ^{n} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {-\sqrt {-{\lambda 2}^{2}{y}^{2}+1}a{\it c2}\,\lambda 1+ \left ( -b{\it c1}\,\sqrt {-{\lambda 1}^{2}{x}^{2}+1}+\lambda 1\, \left ( a\arccos \left ( \lambda 2\,y \right ) y{\it c2}+bx{\it c1}\,\arccos \left ( \lambda 1\,x \right ) \right ) \right ) \lambda 2}{a\lambda 1\,\lambda 2\,b}}}} \]

____________________________________________________________________________________

144.4 Problem 4

Problem Chapter 5.7.2.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu x) w + p \arccos ^n(\beta y) \]

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*ArcCos[mu*x]^m*D[w[x, y], y] == c*ArcCos[nu*x]^k*w[x,y]+p*ArcCos[beta*y]^n; 
 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 :=  a*diff(w(x,y),x)+ b*arccos(mu*x)^m*diff(w(x,y),y) = c*arccos(nu*x)^k*w(x,y)+p*arccos(beta*y)^n; 
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 {p}{a} \left ( \arccos \left ( {\frac {\beta }{\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) } \left ( m+2 \right ) a\mu } \left ( {\frac {\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }b \left ( \left ( m+2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \mu \,x \right ) }}}+ \left ( \left ( -m-2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) +\arccos \left ( \mu \,{\it \_a} \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) - \left ( \arccos \left ( \mu \,{\it \_a} \right ) \right ) ^{m+3/2} \right ) b\sqrt {-{{\it \_a}}^{2}{\mu }^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -\sqrt {\arccos \left ( \mu \,x \right ) }\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }bx\LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) +a\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }y+\arccos \left ( \mu \,{\it \_a} \right ) b{\it \_a}\,\LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) \right ) \right ) } \right ) \right ) ^{n}{{\rm e}^{-{\frac { \left ( \left ( \left ( -2-k \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,{\it \_a} \right ) \right ) +\arccos \left ( \nu \,{\it \_a} \right ) \LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \nu \,{\it \_a} \right ) \right ) - \left ( \arccos \left ( \nu \,{\it \_a} \right ) \right ) ^{3/2+k} \right ) \sqrt {-{{\it \_a}}^{2}{\nu }^{2}+1}+\LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,{\it \_a} \right ) \right ) \arccos \left ( \nu \,{\it \_a} \right ) \nu \,{\it \_a}\, \left ( k+2 \right ) \right ) c}{\sqrt {\arccos \left ( \nu \,{\it \_a} \right ) }a\nu \, \left ( k+2 \right ) }}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac { \left ( \left ( m+2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) b\sqrt {-{\mu }^{2}{x}^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -bx\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) +\sqrt {\arccos \left ( \mu \,x \right ) }ay \right ) }{a\mu \, \left ( m+2 \right ) \sqrt {\arccos \left ( \mu \,x \right ) }}} \right ) \right ) {{\rm e}^{-{\frac {{2}^{k}c{2}^{-k} \left ( \left ( \left ( k+2 \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,x \right ) \right ) -\LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \nu \,x \right ) \right ) \arccos \left ( \nu \,x \right ) + \left ( \arccos \left ( \nu \,x \right ) \right ) ^{3/2+k} \right ) \sqrt {-{\nu }^{2}{x}^{2}+1}-\LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \nu \,x \right ) \right ) \arccos \left ( \nu \,x \right ) \nu \,x \left ( k+2 \right ) \right ) }{a\nu \, \left ( k+2 \right ) \sqrt {\arccos \left ( \nu \,x \right ) }}}}} \]

____________________________________________________________________________________

144.5 Problem 5

Problem Chapter 5.7.2.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu y) w + p \arccos ^n(\beta x) \]

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*ArcCos[mu*x]^m*D[w[x, y], y] == c*ArcCos[nu*y]^k*w[x,y]+p*ArcCos[beta*x]^n; 
 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 :=  a*diff(w(x,y),x)+ b*arccos(mu*x)^m*diff(w(x,y),y) = c*arccos(nu*y)^k*w(x,y)+p*arccos(beta*x)^n; 
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 {p \left ( \arccos \left ( \beta \,{\it \_b} \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \arccos \left ( {\frac {\nu }{\sqrt {\arccos \left ( {\it \_b}\,\mu \right ) } \left ( m+2 \right ) a\mu } \left ( {\frac {\sqrt {\arccos \left ( {\it \_b}\,\mu \right ) }b \left ( \left ( m+2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \mu \,x \right ) }}}+ \left ( \left ( -m-2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( {\it \_b}\,\mu \right ) \right ) +\arccos \left ( {\it \_b}\,\mu \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( {\it \_b}\,\mu \right ) \right ) - \left ( \arccos \left ( {\it \_b}\,\mu \right ) \right ) ^{m+3/2} \right ) b\sqrt {-{{\it \_b}}^{2}{\mu }^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -\sqrt {\arccos \left ( \mu \,x \right ) }\sqrt {\arccos \left ( {\it \_b}\,\mu \right ) }bx\LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) +a\sqrt {\arccos \left ( {\it \_b}\,\mu \right ) }y+\arccos \left ( {\it \_b}\,\mu \right ) b{\it \_b}\,\LommelS 1 \left ( m+1/2,1/2,\arccos \left ( {\it \_b}\,\mu \right ) \right ) \right ) \right ) } \right ) \right ) ^{k}\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( {\frac { \left ( \left ( m+2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) b\sqrt {-{\mu }^{2}{x}^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -bx\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) +\sqrt {\arccos \left ( \mu \,x \right ) }ay \right ) }{a\mu \, \left ( m+2 \right ) \sqrt {\arccos \left ( \mu \,x \right ) }}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( \arccos \left ( {\frac {\nu }{ \left ( m+2 \right ) \mu \,a} \left ( {\frac { \left ( \left ( m+2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) -\arccos \left ( \mu \,x \right ) \LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,x \right ) \right ) + \left ( \arccos \left ( \mu \,x \right ) \right ) ^{m+3/2} \right ) b\sqrt {-{\mu }^{2}{x}^{2}+1}}{\sqrt {\arccos \left ( \mu \,x \right ) }}}-b \left ( {\frac { \left ( m+2 \right ) \LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) }{\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }}}-\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }\LommelS 1 \left ( m+3/2,3/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) + \left ( \arccos \left ( \mu \,{\it \_a} \right ) \right ) ^{m+1} \right ) \sqrt {-{{\it \_a}}^{2}{\mu }^{2}+1}+\mu \, \left ( m+2 \right ) \left ( -\sqrt {\arccos \left ( \mu \,x \right ) }bx\LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,x \right ) \right ) +ya+\sqrt {\arccos \left ( \mu \,{\it \_a} \right ) }b{\it \_a}\,\LommelS 1 \left ( m+1/2,1/2,\arccos \left ( \mu \,{\it \_a} \right ) \right ) \right ) \right ) } \right ) \right ) ^{k}}{d{\it \_a}}}} \]

144 HFOPDE, chapter 5.7.2

144.1 Problem 1

144.2 Problem 2

144.3 Problem 3

144.4 Problem 4

144.5 Problem 5