143 HFOPDE, chapter 5.7.1

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143.1 Problem 1

problem number 1132

Added April 13, 2019.

Problem Chapter 5.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \arcsin ^k(\lambda x) + c_2 \arcsin ^n(\beta y) \]

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*D[w[x, y], y] == w[x,y]+ c1*ArcSin[lambda*x]^k+c2*ArcSin[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*arcsin(lambda*x)^k+c2*arcsin(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;
 

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

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143.2 Problem 2

problem number 1133

Added April 13, 2019.

Problem Chapter 5.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + \arcsin ^k(\lambda x) \arcsin ^n(\beta y) \]

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*D[w[x, y], y] == c*w[x,y]+ ArcSin[lambda*x]^k*ArcSin[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)+ arcsin(lambda*x)^k*arcsin(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;
 

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

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143.3 Problem 3

problem number 1134

Added April 13, 2019.

Problem Chapter 5.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcSin[lambda1*x] + c2*ArcSin[lambda2*y])*w[x,y]+ s1*ArcSin[beta1*x]^n+ s2*ArcSin[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*arcsin(lambda1*x) + c2*arcsin(lambda2*y))*w(x,y)+ s1*arcsin(beta1*x)^n+ s2*arcsin(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;
 

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

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143.4 Problem 4

problem number 1135

Added April 13, 2019.

Problem Chapter 5.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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143.5 Problem 5

problem number 1136

Added April 13, 2019.

Problem Chapter 5.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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