146 HFOPDE, chapter 5.7.4

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146.1 Problem 1

problem number 1147

Added April 13, 2019.

Problem Chapter 5.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcCot[lambda*x]^k+c2*ArcCot[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*arccot(lambda*x)^k+c2*arccot(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;
 

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

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146.2 Problem 2

problem number 1148

Added April 13, 2019.

Problem Chapter 5.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]+ ArcCot[lambda*x]^k*ArcCot[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)+ arccot(lambda*x)^k*arccot(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;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \pi /2-\arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}}{a} \left ( \pi /2-\arctan \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}} \]

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146.3 Problem 3

problem number 1149

Added April 13, 2019.

Problem Chapter 5.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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146.4 Problem 4

problem number 1150

Added April 13, 2019.

Problem Chapter 5.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcCot[mu*x]^m*D[w[x, y], y] == c*ArcCot[nu*x]^k*w[x,y]+p*ArcCot[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*arccot(mu*x)^m*diff(w(x,y),y) = c*arccot(nu*x)^k*w(x,y)+p*arccot(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;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p}{a} \left ( \pi /2-\arctan \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \pi /2-\arctan \left ( {\it \_f}\,\mu \right ) \right ) ^{m}\,{\rm d}{\it \_f}+ \left ( -\int \!{\frac {b \left ( \pi /2-\arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{n}{{\rm e}^{-{\frac {c\int \! \left ( \pi /2-\arctan \left ( \nu \,{\it \_f} \right ) \right ) ^{k}\,{\rm d}{\it \_f}}{a}}}}}{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \pi /2-\arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \pi /2-\arctan \left ( \nu \,x \right ) \right ) ^{k}c}{a}}\,{\rm d}x}} \]

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146.5 Problem 5

problem number 1151

Added April 13, 2019.

Problem Chapter 5.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*ArcCot[mu*x]^m*D[w[x, y], y] == c*ArcCot[nu*y]^k*w[x,y]+p*ArcCot[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*arccot(mu*x)^m*diff(w(x,y),y) = c*arccot(nu*y)^k*w(x,y)+p*arccot(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;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \pi /2-\arctan \left ( \beta \,{\it \_f} \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \pi /2-\arctan \left ( {\frac {\nu }{a} \left ( b\int \! \left ( \pi /2-\arctan \left ( {\it \_f}\,\mu \right ) \right ) ^{m}\,{\rm d}{\it \_f}+ \left ( -\int \!{\frac {b \left ( \pi /2-\arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{k}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \pi /2-\arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( \pi /2-\arctan \left ( \nu \, \left ( \int \!{\frac {b \left ( \pi /2-\arctan \left ( {\it \_b}\,\mu \right ) \right ) ^{m}}{a}}\,{\rm d}{\it \_b}-\int \!{\frac {b \left ( \pi /2-\arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) \right ) \right ) ^{k}}{d{\it \_b}}}} \]