141 HFOPDE, chapter 5.6.4

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141.1 Problem 1

problem number 1118

Added April 11, 2019.

Problem Chapter 5.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{w}_y=cw+k\cot (\lambda x+\mu y)$$

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

$$\left\{\{w(x,y)\to -\frac{k(-2(2a\lambda +2b\mu +ic)e^{\frac{2i\mu (ay-bx)}{a}}\text{Hypergeometric2F1}(1,\frac{ic}{2(a\lambda +b\mu )},\frac{2a\lambda +2b\mu +ic}{2a\lambda +2b\mu },e^{2i(\lambda x+\mu y)})+2ic{e}^{2i(\lambda x+\mu y)}\text{Hypergeometric2F1}(1,1+\frac{ic}{2(a\lambda +b\mu )},2+\frac{ic}{2(a\lambda +b\mu )},e^{2i(\lambda x+\mu y)})+(2a\lambda +2b\mu +ic)(1+e^{\frac{2i\mu (ay-bx)}{a}}))+c(c-2i(a\lambda +b\mu ))e^{\frac{x(c-2ib\mu )}{a}}{c}_1(y-\frac{bx}{a})(e^{\frac{2ib\mu x}{a}}-e^{2i\mu y})}{c(c-2i(a\lambda +b\mu ))(-1+e^{\frac{2i\mu (ay-bx)}{a}})}\}\right\}$$

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)+k*cot(lambda*x+mu*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;
 

$$w(x,y)={\mathrm{e}}^{\frac{cx}{a}}({\int }^x\frac{k}{a}\cot \left(\frac{(\lambda \mathit{\_}\mathit{a}+\mu y)a-b\mu (x-\mathit{\_}\mathit{a})}{a}\right){\mathrm{e}}^{-\frac{\mathit{\_}\mathit{a}c}{a}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{ya-bx}{a}\right))$$

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141.2 Problem 2

problem number 1119

Added April 11, 2019.

Problem Chapter 5.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{w}_y=w+c_1{\cot }^k(\lambda x)+c_2{\cot }^n(\beta y)$$

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*Cot[lambda*x]^k + c2*Cot[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*cot(lambda*x)^k + c2*cot(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;
 

$$w(x,y)={\mathrm{e}}^{\frac{x}{a}}({\int }^x\frac{1}{a}{\mathrm{e}}^{-\frac{\mathit{\_}\mathit{a}}{a}}(\mathit{c}\mathit{1}{(\cot \left(\lambda \mathit{\_}\mathit{a}\right))}^k+\mathit{c}\mathit{2}{(\cot \left(\frac{\beta (ya-b(x-\mathit{\_}\mathit{a}))}{a}\right))}^n)d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{ya-bx}{a}\right))$$

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141.3 Problem 3

problem number 1120

Added April 11, 2019.

Problem Chapter 5.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b{w}_y=cw+{\cot }^k(\lambda x){\cot }^n(\beta y)$$

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]+ Cot[lambda*x]^k * Cot[beta*y]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

$$\left\{\{w(x,y)\to e^{\frac{cx}{a}}({\int }_1^x\frac{e^{-\frac{cK[1]}{a}}{\cot }^k(\lambda K[1]){\cot }^n\left(\beta (\frac{b(K[1]-x)}{a}+y)\right)}{a}dK[1]+c_1(y-\frac{bx}{a}))\}\right\}$$

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)+ cot(lambda*x)^k *cot(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;
 

$$w(x,y)={\mathrm{e}}^{\frac{cx}{a}}({\int }^x\frac{{(\cot \left(\lambda \mathit{\_}\mathit{a}\right))}^k}{a}{(\cot \left(\frac{\beta (ya-b(x-\mathit{\_}\mathit{a}))}{a}\right))}^n{\mathrm{e}}^{-\frac{\mathit{\_}\mathit{a}c}{a}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(\frac{ya-bx}{a}\right))$$

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141.4 Problem 4

problem number 1121

Added April 11, 2019.

Problem Chapter 5.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{w}_x+b\cot (\mu y)w_y=c\cot (\lambda x)w+k\cot (\nu x)$$

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*Cot[mu*y]*D[w[x, y], y] == c*Cot[lambda*x]*w[x,y]+k*Cot[nu*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

$$\{\{w(x,y)\to {\sin }^{\frac{c}{a\lambda }}(\lambda x)({\int }_1^x\frac{k\cot (\nu K[1]){\sin }^{-\frac{c}{a\lambda }}(\lambda K[1])}{a}dK[1]+c_1(\frac{\log (\sec (\mu y))}{\mu }-\frac{bx}{a}))\},\{w(x,y)\to {\sin }^{\frac{c}{a\lambda }}(\lambda x)({\int }_1^x\frac{k\cot (\nu K[2]){\sin }^{-\frac{c}{a\lambda }}(\lambda K[2])}{a}dK[2]+c_1(\frac{\log (\sec (\mu y))}{\mu }-\frac{bx}{a}))\}\}$$

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*cot(mu*y)*diff(w(x,y),y) = c*cot(lambda*x)*w(x,y)+ k*cot(nu*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;
 

$$w(x,y)={(\sin \left(\lambda x\right))}^{\frac{c}{a\lambda }}(-{\int }^y\frac{k}{b}{(\sin \left(\frac{\lambda }{b\mu }(b\mu x+a\ln (\cot \left(\mu y\right))-1/2a\ln ({(\cot \left(\mu y\right))}^2+1)+1/2a\ln (-2{(-1+\cos \left(2\mu \mathit{\_}\mathit{a}\right))}^{-1})-a\ln \left(\frac{\cos \left(\mu \mathit{\_}\mathit{a}\right)}{\sin \left(\mu \mathit{\_}\mathit{a}\right)}\right))\right))}^{-\frac{c}{a\lambda }}(\sin \left(1/2\frac{1}{b\mu }(-\ln (-2{(-1+\cos \left(2\mu \mathit{\_}\mathit{a}\right))}^{-1})a\nu +2\ln \left(\frac{\cos \left(\mu \mathit{\_}\mathit{a}\right)}{\sin \left(\mu \mathit{\_}\mathit{a}\right)}\right)a\nu +\ln ({(\cot \left(\mu y\right))}^2+1)a\nu -2\ln (\cot \left(\mu y\right))a\nu -2b\mu (-\mu \mathit{\_}\mathit{a}+\nu x))\right)-\sin \left(1/2\frac{1}{b\mu }(-\ln (-2{(-1+\cos \left(2\mu \mathit{\_}\mathit{a}\right))}^{-1})a\nu +2\ln \left(\frac{\cos \left(\mu \mathit{\_}\mathit{a}\right)}{\sin \left(\mu \mathit{\_}\mathit{a}\right)}\right)a\nu +\ln ({(\cot \left(\mu y\right))}^2+1)a\nu -2\ln (\cot \left(\mu y\right))a\nu -2b\mu (\mu \mathit{\_}\mathit{a}+\nu x))\right)){(\sin \left(1/2\frac{1}{b\mu }(-\ln (-2{(-1+\cos \left(2\mu \mathit{\_}\mathit{a}\right))}^{-1})a\nu +2\ln \left(\frac{\cos \left(\mu \mathit{\_}\mathit{a}\right)}{\sin \left(\mu \mathit{\_}\mathit{a}\right)}\right)a\nu +\ln ({(\cot \left(\mu y\right))}^2+1)a\nu -2\ln (\cot \left(\mu y\right))a\nu -2b\mu (-\mu \mathit{\_}\mathit{a}+\nu x))\right)+\sin \left(1/2\frac{1}{b\mu }(-\ln (-2{(-1+\cos \left(2\mu \mathit{\_}\mathit{a}\right))}^{-1})a\nu +2\ln \left(\frac{\cos \left(\mu \mathit{\_}\mathit{a}\right)}{\sin \left(\mu \mathit{\_}\mathit{a}\right)}\right)a\nu +\ln ({(\cot \left(\mu y\right))}^2+1)a\nu -2\ln (\cot \left(\mu y\right))a\nu -2b\mu (\mu \mathit{\_}\mathit{a}+\nu x))\right))}^{-1}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(1/2\frac{2b\mu x-a\ln ({(\cot \left(\mu y\right))}^2+1)+2a\ln (\cot \left(\mu y\right))}{b\mu }\right))$$

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141.5 Problem 5

problem number 1122

Added April 11, 2019.

Problem Chapter 5.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$ax{w}_x+by{w}_y=cw+k\cot (\lambda x+\nu y)$$

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

$$\left\{\{w(x,y)\to x^{\frac{c}{a}}({\int }_1^x\frac{kK[1{]}^{-\frac{a+c}{a}}\cot (\nu y{x}^{-\frac{b}{a}}K[1{]}^{\frac{b}{a}}+\lambda K[1])}{a}dK[1]+c_1\left(y{x}^{-\frac{b}{a}}\right))\}\right\}$$

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)+k*cot(lambda*x+nu*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;
 

$$w(x,y)=x^{\frac{c}{a}}({\int }^x\frac{k}{a}\cot (\lambda \mathit{\_}\mathit{a}+\nu y{x}^{-\frac{b}{a}}{\mathit{\_}\mathit{a}}^{\frac{b}{a}}){\mathit{\_}\mathit{a}}^{\frac{-a-c}{a}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{F}\mathit{1}\left(y{x}^{-\frac{b}{a}}\right))$$

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141.6 Problem 6

problem number 1123

Added April 11, 2019.

Problem Chapter 5.6.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{\cot }^n(\lambda x)w_x+b{\cot }^m(\mu x)w_y=c{\cot }^k(\nu x)w+p{\cot }^s(\beta y)$$

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*Cot[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Cot[nu*x]^k*w[x,y]+p*Cot[beta*y]^s; 
 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*cot(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) =c*cot(nu*x)^k*w(x,y)+p*cot(beta*y)^s; 
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;
 

$$w(x,y)={\mathrm{e}}^{\int \frac{{(\cot \left(\nu x\right))}^{k}c{(\cot \left(\lambda x\right))}^{-n}}{a}\mathrm{d}x}(\mathit{\_}\mathit{F}\mathit{1}\left(\frac{1}{a}(ya-b\int {\left(\frac{\cos \left(\mu x\right)}{\sin \left(\mu x\right)}\right)}^m{\left(\frac{\cos \left(\lambda x\right)}{\sin \left(\lambda x\right)}\right)}^{-n}\mathrm{d}x)\right)+{\int }^x\frac{p}{a}{\left(1\cos \left(\frac{\beta }{a}(b\int {\left(\frac{\cos \left(\mathit{\_}\mathit{f}\mu \right)}{\sin \left(\mathit{\_}\mathit{f}\mu \right)}\right)}^m{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}\mathrm{d}\mathit{\_}\mathit{f}+ya-b\int {\left(\frac{\cos \left(\mu x\right)}{\sin \left(\mu x\right)}\right)}^m{\left(\frac{\cos \left(\lambda x\right)}{\sin \left(\lambda x\right)}\right)}^{-n}\mathrm{d}x)\right){(\sin \left(\frac{\beta }{a}(b\int {\left(\frac{\cos \left(\mathit{\_}\mathit{f}\mu \right)}{\sin \left(\mathit{\_}\mathit{f}\mu \right)}\right)}^m{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}\mathrm{d}\mathit{\_}\mathit{f}+ya-b\int {\left(\frac{\cos \left(\mu x\right)}{\sin \left(\mu x\right)}\right)}^m{\left(\frac{\cos \left(\lambda x\right)}{\sin \left(\lambda x\right)}\right)}^{-n}\mathrm{d}x)\right))}^{-1}\right)}^s{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}{\mathrm{e}}^{-\frac{c}{a}\int {\left(\frac{\cos \left(\nu \mathit{\_}\mathit{f}\right)}{\sin \left(\nu \mathit{\_}\mathit{f}\right)}\right)}^k{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}\mathrm{d}\mathit{\_}\mathit{f}}d\mathit{\_}\mathit{f})$$

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141.7 Problem 7

problem number 1124

Added April 11, 2019.

Problem Chapter 5.6.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$a{\cot }^n(\lambda x)w_x+b{\cot }^m(\mu x)w_y=c{\cot }^k(\nu y)w+p{\cot }^s(\beta x)$$

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*Cot[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Cot[nu*y]^k*w[x,y]+p*Cot[beta*x]^s; 
 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*cot(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) =c*cot(nu*y)^k*w(x,y)+p*cot(beta*x)^s; 
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(x,y)={\mathrm{e}}^{{\int }^x\frac{c{(\cot \left(\mathit{\_}\mathit{b}\lambda \right))}^{-n}}{a}{(\cot \left(\frac{\nu }{a}(-b\int {\left(\frac{\cos \left(\mu x\right)}{\sin \left(\mu x\right)}\right)}^m{\left(\frac{\cos \left(\lambda x\right)}{\sin \left(\lambda x\right)}\right)}^{-n}\mathrm{d}x+a(\int \frac{b{(\cot \left(\mathit{\_}\mathit{b}\mu \right))}^m{(\cot \left(\mathit{\_}\mathit{b}\lambda \right))}^{-n}}{a}\mathrm{d}\mathit{\_}\mathit{b}+y))\right))}^{k}d\mathit{\_}\mathit{b}}(\mathit{\_}\mathit{F}\mathit{1}\left(\frac{1}{a}(ya-b\int {\left(\frac{\cos \left(\mu x\right)}{\sin \left(\mu x\right)}\right)}^m{\left(\frac{\cos \left(\lambda x\right)}{\sin \left(\lambda x\right)}\right)}^{-n}\mathrm{d}x)\right)+{\int }^x\frac{p}{a}{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}{\left(\frac{\cos \left(\beta \mathit{\_}\mathit{f}\right)}{\sin \left(\beta \mathit{\_}\mathit{f}\right)}\right)}^s{\mathrm{e}}^{-\frac{c}{a}\int {\left(1\cos \left(\frac{\nu }{a}(b\int {\left(\frac{\cos \left(\mathit{\_}\mathit{f}\mu \right)}{\sin \left(\mathit{\_}\mathit{f}\mu \right)}\right)}^m{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}\mathrm{d}\mathit{\_}\mathit{f}+ya-b\int {\left(\frac{\cos \left(\mu x\right)}{\sin \left(\mu x\right)}\right)}^m{\left(\frac{\cos \left(\lambda x\right)}{\sin \left(\lambda x\right)}\right)}^{-n}\mathrm{d}x)\right){(\sin \left(\frac{\nu }{a}(b\int {\left(\frac{\cos \left(\mathit{\_}\mathit{f}\mu \right)}{\sin \left(\mathit{\_}\mathit{f}\mu \right)}\right)}^m{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}\mathrm{d}\mathit{\_}\mathit{f}+ya-b\int {\left(\frac{\cos \left(\mu x\right)}{\sin \left(\mu x\right)}\right)}^m{\left(\frac{\cos \left(\lambda x\right)}{\sin \left(\lambda x\right)}\right)}^{-n}\mathrm{d}x)\right))}^{-1}\right)}^k{\left(\frac{\cos \left(\lambda \mathit{\_}\mathit{f}\right)}{\sin \left(\lambda \mathit{\_}\mathit{f}\right)}\right)}^{-n}\mathrm{d}\mathit{\_}\mathit{f}}d\mathit{\_}\mathit{f})$$