7.2.17 6.3

7.2.17.1 [667] problem number 1

problem number 667

Added January 14, 2019.

Problem 2.6.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(-ax-bx+\frac{\log (\cos (\lambda x))}{\lambda }+y)\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{((-2a-2b)x+2y)\lambda -\ln ({\tan }^2\left(\lambda x\right)+1)}{2\lambda }\right)$$

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7.2.17.2 [668] problem number 2

problem number 668

Added January 14, 2019.

Problem 2.6.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(-x+\frac{-i(a+b-i)\log (-\tan (\lambda y)+i)+i(a+b+i)\log (\tan (\lambda y)+i)+2\log (a+b+\tan (\lambda y))}{2\lambda (a+b-i)(a+b+i)})\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{(2a^{2}x+4abx+2b^{2}x+2x)\lambda +(-2a-2b)\arctan (\tan \left(\lambda y\right))+\ln ({\tan }^2\left(\lambda y\right)+1)-2\ln (a+b+\tan \left(\lambda y\right))}{2(a^2+2ab+b^2+1)\lambda }\right)$$

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7.2.17.3 [669] problem number 3

problem number 669

Added January 14, 2019.

Problem 2.6.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(\frac{{\tan }^{1-n}(\mu y){}_2{F}_1(1,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};-{\tan }^2(\mu y))}{\mu -\mu n}-\frac{a{\tan }^{k+1}(\lambda x){}_2{F}_1(1,\frac{k+1}{2};\frac{k+3}{2};-{\tan }^2(\lambda x))}{k\lambda +\lambda })\}\right\}$$

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a *tan(lambda*x)^k * tan(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'));
 

$$w(x,y)=\mathit{\text{\_F1}}(-(\int ({\tan }^k\left(\lambda x\right))dx)+\int \frac{{\tan }^{-n}\left(\mu y\right)}{a}dy)$$ Has unresolved integrals

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7.2.17.4 [670] problem number 4

problem number 670

Added January 14, 2019.

Problem 2.6.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + a*lambda + a*(lambda - a)*Tan[lambda*x]^2)*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+ a *lambda + a*(lambda -a) *tan(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{2(\cos \left(2\lambda x\right)+1)\lambda \mathrm{LegendreP}(\frac{2a+\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))+(-4y({\cos }^3\left(\lambda x\right))-a\sin \left(\lambda x\right)-a\sin \left(3\lambda x\right))\mathrm{LegendreP}(\frac{2a-\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))}{-2(\cos \left(2\lambda x\right)+1)\lambda \mathrm{LegendreQ}(\frac{2a+\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))+(4y({\cos }^3\left(\lambda x\right))+a\sin \left(\lambda x\right)+a\sin \left(3\lambda x\right))\mathrm{LegendreQ}(\frac{2a-\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))}\right)$$

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7.2.17.5 [671] problem number 5

problem number 671

Added January 14, 2019.

Problem 2.6.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(y^2+{\lambda }^2+3a\lambda +a(\lambda -a){\tan }^2(\lambda x))w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda^2 + 3*a*lambda + a*(lambda - a)*Tan[lambda*x]^2)*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^2 +3*a*lambda +a*(lambda-a)*tan(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{4\lambda \mathrm{LegendreP}(\frac{2a+3\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))({\cos }^2\left(\lambda x\right))-2(\frac{\lambda ({\sin }^3\left(\lambda x\right))}{2}+(a+\frac{3\lambda }{2})({\cos }^2\left(\lambda x\right))\sin \left(\lambda x\right)-\frac{\lambda \sin \left(\lambda x\right)}{2}+(-y({\sin }^2\left(\lambda x\right))+y)\cos \left(\lambda x\right))\mathrm{LegendreP}(\frac{2a+\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))}{-4\lambda \mathrm{LegendreQ}(\frac{2a+3\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))({\cos }^2\left(\lambda x\right))+2(\frac{\lambda ({\sin }^3\left(\lambda x\right))}{2}+(a+\frac{3\lambda }{2})({\cos }^2\left(\lambda x\right))\sin \left(\lambda x\right)-\frac{\lambda \sin \left(\lambda x\right)}{2}+(-y({\sin }^2\left(\lambda x\right))+y)\cos \left(\lambda x\right))\mathrm{LegendreQ}(\frac{2a+\lambda }{2\lambda },\frac{2a-\lambda }{2\lambda },\sin \left(\lambda x\right))}\right)$$

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7.2.17.6 [672] problem number 6

problem number 672

Added January 14, 2019.

Problem 2.6.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(y^2+ax{\tan }^k(bx)y+a{\tan }^k(bx))w_y=0$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(-\frac{\exp (-{\int }_1^x-aK[5]{\tan }^k(bK[5])dK[5])}{x^{2}y+x}-{\int }_1^x\frac{\exp (-{\int }_1^{K[6]}-aK[5]{\tan }^k(bK[5])dK[5])}{K[6{]}^2}dK[6])\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{xy(\int {\mathrm{e}}^{\int \frac{a{x}^2({\tan }^k\left(bx\right))-2}{x}dx}dx)+x{\mathrm{e}}^{\int \frac{a{x}^2({\tan }^k\left(bx\right))-2}{x}dx}+\int {\mathrm{e}}^{\int \frac{a{x}^2({\tan }^k\left(bx\right))-2}{x}dx}dx}{xy+1}\right)$$

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7.2.17.7 [673] problem number 7

problem number 673

Added January 14, 2019.

Problem 2.6.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x-((k+1)x^k{y}^2-a{x}^{k+1}(\tan x{)}^{m}y+a(\tan x{)}^m)w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - a*x^(k + 1)*Tan[x]^m*y + a*Tan[x]^m)*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- a*x^(k+1)*tan(x)^m*y + a*tan(x)^m )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{-x^{k+1}{\mathrm{e}}^{\int \frac{ax{x}^{k+1}({\tan }^m\left(x\right))-2k-2}{x}dx}+(y{x}^{k+1}-1)(k+1)(\int \frac{x^{-k}{\mathrm{e}}^{a(\int x^{k+1}{(-\frac{i({\mathrm{e}}^{2ix}-1)}{{\mathrm{e}}^{2ix}+1})}^{m}dx)}}{x^2}dx)}{y{x}^{k+1}-1}\right)$$

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7.2.17.8 [674] problem number 8

problem number 674

Added January 20, 2019.

Problem 2.6.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Tan[lambda*x]^n*y^2 - a*b^2*Tan[lambda*x]^(n + 2) + b*lambda*Tan[lambda*x]^2 + b*lambda)*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)+(a*tan(lambda*x)^n*y^2- a*b^2*tan(lambda*x)^(n+2) + b*lambda*tan(lambda*x)^2+ b*lambda)*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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7.2.17.9 [675] problem number 9

problem number 675

Added January 20, 2019.

Problem 2.6.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(a{\tan }^k(\lambda x+\mu )(y-b{x}^n-c{)}^2+y-b{x}^n+bn{x}^{n-1}-c)w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Tan[lambda*x + mu]^k*(y - b*x^n - c)^2 + y - b*x^n + b*n*x^(n - 1) - c)*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)+(a *tan(lambda*x+mu)^k*(y-b*x^n-c)^2 + y- b*x^n + b*n*x^(n-1)-c)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

time expired

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7.2.17.10 [676] problem number 10

problem number 676

Added January 20, 2019.

Problem 2.6.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}(ab(\int x^{k-1}({\tan }^m\left(\lambda x\right))dx)-\arctan \left(\frac{y{x}^{-k}}{b}\right))$$

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7.2.17.11 [677] problem number 11

problem number 677

Added January 20, 2019.

Problem 2.6.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$(a\tan (\lambda x)+b)w_x+(y^2+c\tan (\mu x)y-k^2+ck\tan (\mu x))w_y=0$$

Mathematica ✗

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

Failed

Maple ✓

restart; 
pde :=   (a*tan(lambda*x)+b)*diff(w(x,y),x)+ (y^2+ c *tan(mu*x)*y - k^2 + c*k*tan(mu*x) )*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{\text{\_F1}}\left(\frac{((k+y){({\mathrm{e}}^{2i\mu x}+1)}^{\frac{i{a}^{2}c}{(ib+a)(a^2+b^2)\mu }}{({\mathrm{e}}^{2i\mu x}+1)}^{\frac{i{b}^{2}c}{(ib+a)(a^2+b^2)\mu }}(\int \frac{{\left(\frac{a\sin \left(\lambda x\right)+b\cos \left(\lambda x\right)}{\cos \left(\lambda x\right)}\right)}^{-\frac{2ak}{(a^2+b^2)\lambda }}{\left(\frac{2}{\cos \left(2\lambda x\right)+1}\right)}^{\frac{ak}{(a^2+b^2)\lambda }}{({\mathrm{e}}^{2i\mu x}+1)}^{-\frac{i{a}^{2}c}{(ib+a)(a^2+b^2)\mu }}{({\mathrm{e}}^{2i\mu x}+1)}^{-\frac{i{b}^{2}c}{(ib+a)(a^2+b^2)\mu }}\cos \left(\lambda x\right){\mathrm{e}}^{-\frac{i{a}^{2}c(\int \frac{4i({\mathrm{e}}^{2i\mu x}-1)a}{(ib+a)(-a+ib+(ib+a){\mathrm{e}}^{2i\lambda x})({\mathrm{e}}^{2i\mu x}+1)}dx)}{2a^2+2b^2}}{\mathrm{e}}^{-\frac{i{b}^{2}c(\int \frac{4i({\mathrm{e}}^{2i\mu x}-1)a}{(ib+a)(-a+ib+(ib+a){\mathrm{e}}^{2i\lambda x})({\mathrm{e}}^{2i\mu x}+1)}dx)}{2a^2+2b^2}}{\mathrm{e}}^{-\frac{2bk\arctan \left(\frac{\sin \left(\lambda x\right)}{\cos \left(\lambda x\right)}\right)}{(a^2+b^2)\lambda }}{\mathrm{e}}^{-\frac{a^{2}cx}{(a^2+b^2)(ib+a)}}{\mathrm{e}}^{-\frac{b^{2}cx}{(a^2+b^2)(ib+a)}}}{a\sin \left(\lambda x\right)+b\cos \left(\lambda x\right)}dx){\mathrm{e}}^{\frac{a^{2}cx}{(a^2+b^2)(ib+a)}}{\mathrm{e}}^{\frac{b^{2}cx}{(a^2+b^2)(ib+a)}}{\mathrm{e}}^{\frac{2bk\arctan \left(\frac{\sin \left(\lambda x\right)}{\cos \left(\lambda x\right)}\right)}{(a^2+b^2)\lambda }}+{\left(\frac{a\sin \left(\lambda x\right)+b\cos \left(\lambda x\right)}{\cos \left(\lambda x\right)}\right)}^{-\frac{2ak}{(a^2+b^2)\lambda }}{\left(\frac{2}{\cos \left(2\lambda x\right)+1}\right)}^{\frac{ak}{(a^2+b^2)\lambda }}{\mathrm{e}}^{-\frac{i{a}^{2}c(\int \frac{4i({\mathrm{e}}^{2i\mu x}-1)a}{(ib+a)(-a+ib+(ib+a){\mathrm{e}}^{2i\lambda x})({\mathrm{e}}^{2i\mu x}+1)}dx)}{2a^2+2b^2}}{\mathrm{e}}^{-\frac{i{b}^{2}c(\int \frac{4i({\mathrm{e}}^{2i\mu x}-1)a}{(ib+a)(-a+ib+(ib+a){\mathrm{e}}^{2i\lambda x})({\mathrm{e}}^{2i\mu x}+1)}dx)}{2a^2+2b^2}}){({\mathrm{e}}^{2i\mu x}+1)}^{-\frac{i{a}^{2}c}{(ib+a)(a^2+b^2)\mu }}{({\mathrm{e}}^{2i\mu x}+1)}^{-\frac{i{b}^{2}c}{(ib+a)(a^2+b^2)\mu }}{\mathrm{e}}^{-\frac{2bk\arctan \left(\frac{\sin \left(\lambda x\right)}{\cos \left(\lambda x\right)}\right)}{(a^2+b^2)\lambda }}{\mathrm{e}}^{-\frac{a^{2}cx}{(a^2+b^2)(ib+a)}}{\mathrm{e}}^{-\frac{b^{2}cx}{(a^2+b^2)(ib+a)}}}{k+y}\right)$$

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7.2.17.12 [678] problem number 12

problem number 678

Added January 20, 2019.

Problem 2.6.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

Failed

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}((n-1)a(\int y^m({\tan }^{-k}\left(\lambda y\right)){\mathrm{e}}^{(n-1)b(\int ({\tan }^{-k}\left(\lambda y\right))dy)}dy)+x^{-n+1}{\mathrm{e}}^{(n-1)b(\int ({\tan }^{-k}\left(\lambda y\right))dy)})$$

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7.2.17.13 [679] problem number 13

problem number 679

Added January 20, 2019.

Problem 2.6.3.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$(a{x}^n+bx{\tan }^{m}y)w_x+y^k{w}_y=0$$

Mathematica ✗

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

Failed

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}((n-1)a(\int y^{-k}{\mathrm{e}}^{(n-1)b(\int y^{-k}({\tan }^m\left(y\right))dy)}dy)+x^{-n+1}{\mathrm{e}}^{(n-1)b(\int y^{-k}({\tan }^m\left(y\right))dy)})$$

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7.2.17.14 [680] problem number 14

problem number 680

Added January 20, 2019.

Problem 2.6.3.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

Failed

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}((n-1)a(\int ({\tan }^{-k}\left(\lambda y\right)){\mathrm{e}}^{(n-1)b(\int ({\tan }^m\left(y\right))({\tan }^{-k}\left(\lambda y\right))dy)}dy)+x^{-n+1}{\mathrm{e}}^{(n-1)b(\int ({\tan }^m\left(y\right))({\tan }^{-k}\left(\lambda y\right))dy)})$$

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7.2.17.15 [681] problem number 15

problem number 681

Added January 20, 2019.

Problem 2.6.3.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$(a{x}^n{\tan }^{m}y+bx)w_x+{\tan }^k(\lambda y)w_y=0$$

Mathematica ✗

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

Failed

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}((n-1)a(\int ({\tan }^m\left(y\right))({\tan }^{-k}\left(\lambda y\right)){\mathrm{e}}^{(n-1)b(\int ({\tan }^{-k}\left(\lambda y\right))dy)}dy)+x^{-n+1}{\mathrm{e}}^{(n-1)b(\int ({\tan }^{-k}\left(\lambda y\right))dy)})$$

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