6.2.21 7.2

6.2.21.1 [715] problem number 1

problem number 715

Added January 29, 2019.

Problem 2.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(a{\arccos }^k(\lambda x)+b)w_y=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcCos[lambda*x]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1(-\frac{a{\cos }^{-1}(\lambda x{)}^k{(-i{\cos }^{-1}(\lambda x))}^{-k}\text{Gamma}(k+1,-i{\cos }^{-1}(\lambda x))+a{(i{\cos }^{-1}(\lambda x))}^{-k}{\cos }^{-1}(\lambda x{)}^k\text{Gamma}(k+1,i{\cos }^{-1}(\lambda x))+2b\lambda x-2\lambda y}{2\lambda })\}\right\}$$

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arccos(lambda*x)^k + b   )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-bx+y+\frac{a{2}^k\sqrt{\pi }}{\lambda }(\frac{{(\arccos \left(\lambda x\right))}^{k+1}{2}^{-k}}{\sqrt{\pi }(k+2)}\sqrt{-{\lambda }^2{x}^2+1}-\frac{2^{-k}}{\sqrt{\pi }(k+2)}\sqrt{\arccos \left(\lambda x\right)}\mathrm{LommelS}1(k+\frac{3}{2},\frac{3}{2},\arccos \left(\lambda x\right))\sqrt{-{\lambda }^2{x}^2+1}-3\frac{2^{-1-k}(2/3k+4/3)(\lambda x\arccos \left(\lambda x\right)-\sqrt{-{\lambda }^2{x}^2+1})\mathrm{LommelS}1(k+1/2,1/2,\arccos \left(\lambda x\right))}{\sqrt{\pi }(k+2)\sqrt{\arccos \left(\lambda x\right)}}))$$

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6.2.21.2 [716] problem number 2

problem number 716

Added January 29, 2019.

Problem 2.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(a{\arccos }^k(\lambda y)+b)w_y=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcCos[lambda*y]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arccos(lambda*y)^k + b   )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.2.21.3 [717] problem number 3

problem number 717

Added January 29, 2019.

Problem 2.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+k{\arccos }^n(ax+by+c)w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + k*ArcCos[a*x + b*y + c]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+k*arccos(a*x+b*y+c)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-{\int }^{\frac{ax+by}{b}}{(k{(\arccos (\mathit{\_}\mathit{a}b+c))}^{n}b+a)}^{-1}d\mathit{\_}\mathit{a}b+x)$$

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6.2.21.4 [718] problem number 4

problem number 718

Added January 29, 2019.

Problem 2.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+a{\arccos }^k(\lambda x){\arccos }^n(\mu y)w_y=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*ArcCos[lambda*x]^k*ArcCos[mu*y]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1\left(\frac{{({\cos }^{-1}(\lambda x{)}^2)}^{-k}(-a{(i{\cos }^{-1}(\lambda x))}^k{\cos }^{-1}(\lambda x{)}^k\text{Gamma}(k+1,-i{\cos }^{-1}(\lambda x))-a{(-i{\cos }^{-1}(\lambda x))}^k{\cos }^{-1}(\lambda x{)}^k\text{Gamma}(k+1,i{\cos }^{-1}(\lambda x))+\frac{\lambda {({\cos }^{-1}(\lambda x{)}^2)}^k{\cos }^{-1}(\mu y{)}^{-n}({(-i{\cos }^{-1}(\mu y))}^n\text{Gamma}(1-n,-i{\cos }^{-1}(\mu y))+{(i{\cos }^{-1}(\mu y))}^n\text{Gamma}(1-n,i{\cos }^{-1}(\mu y)))}{\mu })}{2\lambda }\right)\}\right\}$$

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a*arccos(lambda*x)^k*arccos(mu*y)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-\frac{1}{(k+2)\lambda (-2+n)a\mu }(a\sqrt{\arccos \left(\mu y\right)}(-2+n)2^k\mu (-2\frac{2^{-1-k}(k+2)\mathrm{LommelS}1(k+1/2,1/2,\arccos \left(\lambda x\right))}{\sqrt{\arccos \left(\lambda x\right)}}+2^{-k}(\sqrt{\arccos \left(\lambda x\right)}\mathrm{LommelS}1(k+\frac{3}{2},\frac{3}{2},\arccos \left(\lambda x\right))-{(\arccos \left(\lambda x\right))}^{k+1}))\sqrt{-{\lambda }^2{x}^2+1}+2\lambda (k+2)(((-1+n/2)\mathrm{LommelS}1(-n+1/2,1/2,\arccos \left(\mu y\right))+1/2\arccos \left(\mu y\right)\mathrm{LommelS}1(-n+3/2,3/2,\arccos \left(\mu y\right))-1/2{(\arccos \left(\mu y\right))}^{-n+3/2})\sqrt{-{\mu }^2{y}^2+1}+(a{2}^{-1-k}\sqrt{\arccos \left(\lambda x\right)}\sqrt{\arccos \left(\mu y\right)}\mathrm{LommelS}1(k+1/2,1/2,\arccos \left(\lambda x\right))x{2}^k-1/2\arccos \left(\mu y\right)y\mathrm{LommelS}1(-n+1/2,1/2,\arccos \left(\mu y\right)))(-2+n)\mu ))\frac{1}{\sqrt{\arccos \left(\mu y\right)}})$$

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6.2.21.5 [719] problem number 5

problem number 719

Added January 29, 2019.

Problem 2.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(y^2+\lambda (\arccos x{)}^{n}y-a^2+a\lambda (\arccos x{)}^n)w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*ArcCos[x]^n*y - a^2 + a*lambda*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( y^2+lambda*arccos(x)^n*y- a^2 + a*lambda*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{1}{a+y}((-a-y)\int \frac{1}{(2+n)(a+y)}((-n\lambda \arccos \left(x\right)(2+n)\mathrm{LommelS}1(-\frac{1}{2}+n,\frac{1}{2},\arccos \left(x\right))+\lambda (2+n)\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(x\right))-\arccos \left(x\right)\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(x\right))\lambda +{(\arccos \left(x\right))}^{n+\frac{3}{2}}\lambda +{(\arccos \left(x\right))}^{\frac{3}{2}}(2+n)(a+y))\sqrt{-x^2+1}-x(-n{(\arccos \left(x\right))}^2(2+n)\mathrm{LommelS}1(-\frac{1}{2}+n,\frac{1}{2},\arccos \left(x\right))+\arccos \left(x\right)(2+n)\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(x\right))-\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(x\right)){(\arccos \left(x\right))}^2+{(\arccos \left(x\right))}^{n+\frac{5}{2}})\lambda ){\mathrm{e}}^{-2\frac{1/2((2+n)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))-\arccos \left(x\right)\mathrm{LommelS}1(n+3/2,3/2,\arccos \left(x\right))+{(\arccos \left(x\right))}^{n+3/2})\lambda \sqrt{-x^2+1}+x(-1/2\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))\lambda \arccos \left(x\right)+a\sqrt{\arccos \left(x\right)})(2+n)}{(2+n)\sqrt{\arccos \left(x\right)}}}\frac{1}{\sqrt{-x^2+1}}{(\arccos \left(x\right))}^{-\frac{3}{2}}\mathrm{d}x-{\mathrm{e}}^{-2\frac{1/2((2+n)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))-\arccos \left(x\right)\mathrm{LommelS}1(n+3/2,3/2,\arccos \left(x\right))+{(\arccos \left(x\right))}^{n+3/2})\lambda \sqrt{-x^2+1}+x(-1/2\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))\lambda \arccos \left(x\right)+a\sqrt{\arccos \left(x\right)})(2+n)}{(2+n)\sqrt{\arccos \left(x\right)}}})\right)$$

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6.2.21.6 [720] problem number 6

problem number 720

Added January 29, 2019.

Problem 2.7.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(y^2+\lambda x(\arccos x{)}^{n}y+\lambda (\arccos x{)}^n)w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*x*ArcCos[x]^n*y + a*lambda*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( y^2+lambda*x*arccos(x)^n*y + a*lambda*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.21.7 [721] problem number 7

problem number 721

Added January 29, 2019.

Problem 2.7.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x-((k+1)x^k{y}^2-\lambda (\arccos x{)}^n(x^{k+1}y-1))w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcCos[x]^n*(x^(k + 1)*y - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)-( (k+1)*x^k*y^2 -lambda*arccos(x)^n*(x^(k+1)*y-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{1}{x^{k+1}y-1}(-{\mathrm{e}}^{\int \frac{x^{k+1}{(\arccos \left(x\right))}^n\lambda x-2k-2}{x}\mathrm{d}x}{x}^{k+1}+\int \frac{{\mathrm{e}}^{\lambda \int x^{k+1}{(\arccos \left(x\right))}^n\mathrm{d}x}}{x^k{x}^2}\mathrm{d}x(x^{k+1}y-1)(k+1))\right)$$

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6.2.21.8 [722] problem number 8

problem number 722

Added January 29, 2019.

Problem 2.7.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(\lambda (\arccos x{)}^n{y}^2+ay+ab-b^2\lambda (\arccos x{)}^n)w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + a*y + a*b - b^2*lambda*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ a*y+ a*b - b^2*lambda*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{1}{b+y}((-b-y)\int \frac{\lambda }{(2+n)(b+y)}{\mathrm{e}}^{\frac{1}{2+n}(2((2+n)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))-\arccos \left(x\right)\mathrm{LommelS}1(n+3/2,3/2,\arccos \left(x\right))+{(\arccos \left(x\right))}^{n+3/2})\lambda b\sqrt{-x^2+1}+x(2+n)(-2b\lambda \arccos \left(x\right)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))+a\sqrt{\arccos \left(x\right)}))\frac{1}{\sqrt{\arccos \left(x\right)}}}((2bn\arccos \left(x\right)(2+n)\mathrm{LommelS}1(-1/2+n,1/2,\arccos \left(x\right))-2b(2+n)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))+2b\arccos \left(x\right)\mathrm{LommelS}1(n+3/2,3/2,\arccos \left(x\right))+{(\arccos \left(x\right))}^{n+\frac{3}{2}}((b+y)n+2y))\sqrt{-x^2+1}+2bx(-n{(\arccos \left(x\right))}^2(2+n)\mathrm{LommelS}1(-1/2+n,1/2,\arccos \left(x\right))+\arccos \left(x\right)(2+n)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))-\mathrm{LommelS}1(n+3/2,3/2,\arccos \left(x\right)){(\arccos \left(x\right))}^2+{(\arccos \left(x\right))}^{n+5/2})){(\arccos \left(x\right))}^{-\frac{3}{2}}\frac{1}{\sqrt{-x^2+1}}\mathrm{d}x-{\mathrm{e}}^{\frac{1}{2+n}(2((2+n)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))-\arccos \left(x\right)\mathrm{LommelS}1(n+3/2,3/2,\arccos \left(x\right))+{(\arccos \left(x\right))}^{n+3/2})\lambda b\sqrt{-x^2+1}+x(2+n)(-2b\lambda \arccos \left(x\right)\mathrm{LommelS}1(n+1/2,1/2,\arccos \left(x\right))+a\sqrt{\arccos \left(x\right)}))\frac{1}{\sqrt{\arccos \left(x\right)}}})\right)$$

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6.2.21.9 [723] problem number 9

problem number 723

Added January 29, 2019.

Problem 2.7.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(\lambda (\arccos x{)}^n{y}^2-b\lambda x^m(\arccos x{)}^{n}y+bm{x}^{m-1})w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 - b*lambda*x^m*ArcCos[x]^n*y + b*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(x)^n*y^2- b*lambda*x^m*arccos(x)^n*y + b*m*x^(m-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.21.10 [724] problem number 10

problem number 724

Added January 29, 2019.

Problem 2.7.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(\lambda (\arccos x{)}^n{y}^2+bm{x}^{m-1}-\lambda b^2{x}^{2m}(\arccos x{)}^n)w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ b*m*x^(m-1) - lambda*b^2*x^(2*m)*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.21.11 [725] problem number 11

problem number 725

Added January 29, 2019.

Problem 2.7.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(\lambda (\arccos x{)}^n(y-a{x}^m-b{)}^2+am{x}^{m-1})w_y=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCos[x]^n*(y - a*x^m - b)^2 + a*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1\left(\frac{1}{2}(\lambda {(i{\cos }^{-1}(x))}^n{\cos }^{-1}(x{)}^n{({\cos }^{-1}(x{)}^2)}^{-n}\text{Gamma}(n+1,-i{\cos }^{-1}(x))+\lambda {(-i{\cos }^{-1}(x))}^n{\cos }^{-1}(x{)}^n{({\cos }^{-1}(x{)}^2)}^{-n}\text{Gamma}(n+1,i{\cos }^{-1}(x))-\frac{2}{a{x}^m+b-y})\right)\}\right\}$$

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccos(x)^n*(y- a*x^m-b)^2 + a*m*x^(m-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-\frac{1}{(2+n)(a{x}^m+b-y)}(\lambda ((2+n)\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(x\right))+{(\arccos \left(x\right))}^n{(\arccos \left(x\right))}^{\frac{3}{2}}-\arccos \left(x\right)\mathrm{LommelS}1(n+\frac{3}{2},\frac{3}{2},\arccos \left(x\right)))(a{x}^m+b-y)\sqrt{-x^2+1}-(2+n)(x\lambda \arccos \left(x\right)(a{x}^m+b-y)\mathrm{LommelS}1(n+\frac{1}{2},\frac{1}{2},\arccos \left(x\right))-\sqrt{\arccos \left(x\right)}))\frac{1}{\sqrt{\arccos \left(x\right)}})$$

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6.2.21.12 [726] problem number 12

problem number 726

Added January 29, 2019.

Problem 2.7.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$x{w}_x+(\lambda (\arccos x{)}^n{y}^2+ky+\lambda b^2{x}^{2k}(\arccos x{)}^n)w_y=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcCos[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1({\tan }^{-1}\left(\frac{y{x}^{-k}}{\sqrt{b^2}}\right)-\sqrt{b^2}{\int }_1^x\lambda {\cos }^{-1}(K[1]{)}^{n}K[1{]}^{k-1}dK[1])\}\right\}$$

Maple ✓

restart; 
pde :=  x*diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ k*y + lambda*b^2*x^(2*k)*arccos(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(\lambda b\int x^{k-1}{(\arccos \left(x\right))}^n\mathrm{d}x-\arctan \left(\frac{x^{-k}y}{b}\right))$$

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