6.2.24 8.1

6.2.24.1 [751] problem number 1

problem number 751

Added Feb. 4, 2019.

Problem 2.8.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$w_x+(f(x)y+g(x))w_y=0$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(y\exp (-{\int }_1^{x}f(K[1])dK[1])-{\int }_1^x\exp (-{\int }_1^{K[2]}f(K[1])dK[1])g(K[2])dK[2])\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-\int g\left(x\right){\mathrm{e}}^{-\int f\left(x\right)\mathrm{d}x}\mathrm{d}x+y{\mathrm{e}}^{-\int f\left(x\right)\mathrm{d}x})$$

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6.2.24.2 [752] problem number 2

problem number 752

Added Feb. 4, 2019.

Problem 2.8.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y+g(x)y^k)w_y=0$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1((k-1){\int }_1^x\exp ((k-1){\int }_1^{K[2]}f(K[1])dK[1])g(K[2])dK[2]+y^{1-k}\exp ((k-1){\int }_1^{x}f(K[1])dK[1]))\}\right\}$$

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( f(x)*y+g(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'));
 

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}((k-1)\int {\mathrm{e}}^{(k-1)\int f\left(x\right)\mathrm{d}x}g\left(x\right)\mathrm{d}x+y^{-k+1}{\mathrm{e}}^{(k-1)\int f\left(x\right)\mathrm{d}x})$$

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6.2.24.3 [753] problem number 3

problem number 753

Added Feb. 4, 2019.

Problem 2.8.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(y^2+f(x)y-a^2-af(x))w_y=0$$

Mathematica ✗

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{(a-y)\int {\mathrm{e}}^{\int f\left(x\right)\mathrm{d}x+2ax}\mathrm{d}x-{\mathrm{e}}^{\int f\left(x\right)\mathrm{d}x+2ax}}{a-y}\right)$$

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6.2.24.4 [754] problem number 4

problem number 754

Added Feb. 4, 2019.

Problem 2.8.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(y^2+xf(x)y+f(x))w_y=0$$

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.24.5 [755] problem number 5

problem number 755

Added Feb. 4, 2019.

Problem 2.8.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x-((k+1)x^k{y}^2-x^{k+1}f(x)y+f(x))w_y=0$$

Mathematica ✗

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

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

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6.2.24.6 [756] problem number 6

problem number 756

Added Feb. 4, 2019.

Problem 2.8.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y^2+ay-ab-b^{2}f(x))w_y=0$$

Mathematica ✗

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{(b-y)\int {\mathrm{e}}^{ax+2b\int f\left(x\right)\mathrm{d}x}f\left(x\right)\mathrm{d}x-{\mathrm{e}}^{ax+2b\int f\left(x\right)\mathrm{d}x}}{b-y}\right)$$

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6.2.24.7 [757] problem number 7

problem number 757

Added Feb. 4, 2019.

Problem 2.8.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f[x]y^2-a{x}^{n}f[x]y+an{x}^{n-1})w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 - a*x^n*f[x]*y + a*n*x^(n - 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)+( f(x)*y^2-a*x^n*f(x)*y+a*n*x^(n-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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6.2.24.8 [758] problem number 8

problem number 758

Added Feb. 4, 2019.

Problem 2.8.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y^2+an{x}^{n-1}-a^2{x}^{2n}f(x))w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 + a*n*x^(n - 1) - a^2*x^(2*n)*f[x])*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)+(  f(x)*y^2+a*n*x^(n-1)-a^2*x^(2*n)*f(x))*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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6.2.24.9 [759] problem number 9

problem number 759

Added Feb. 4, 2019.

Problem 2.8.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y^2+g(x)y-a^{2}f(x)-ag(x))w_y=0$$

Mathematica ✗

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}\left(\frac{(a-y)\int {\mathrm{e}}^{\int g\left(x\right)\mathrm{d}x+2a\int f\left(x\right)\mathrm{d}x}f\left(x\right)\mathrm{d}x-{\mathrm{e}}^{\int g\left(x\right)\mathrm{d}x+2a\int f\left(x\right)\mathrm{d}x}}{a-y}\right)$$

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6.2.24.10 [760] problem number 10

problem number 760

Added Feb. 4, 2019.

Problem 2.8.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y^2+g(x)y+an{x}^{n-1}-a{x}^{n}g(x)-a^2{x}^{2n}f(x))w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 + g[x]*y + a*n*x^(n - 1) - a*x^n*g[x] - a^2*x^(2*n)*f[x])*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)+(  f(x)*y^2+g(x)*y+a*n*x^(n-1) - a*x^n*g(x)-a^2*x^(2*n)*f(x))*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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6.2.24.11 [761] problem number 11

problem number 761

Added Feb. 4, 2019.

Problem 2.8.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y^2-a{x}^{n}g(x)y+an{x}^{n-1}+a^2{x}^{2n}(g(x)-f(x)))w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 - a*x^n*g*x*y + a*n*x^(n - 1) + a^2*x^(2*n)*(g*x - f*x))*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)+( f(x)*y^2-a*x^n*g(x)*y+a*n*x^(n-1)+a^2*x^(2*n)*(g(x)-f(x)))*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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6.2.24.12 [762] problem number 12

problem number 762

Added Feb. 4, 2019.

Problem 2.8.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$x{w}_x+(f(x)y^2+ny+a{x}^{2n}f(x))w_y=0$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1({\tan }^{-1}\left(\frac{y{x}^{-n}}{\sqrt{a}}\right)-\sqrt{a}{\int }_1^{x}f(K[1])K[1{]}^{n-1}dK[1])\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(\sqrt{a}\int x^{n-1}f\left(x\right)\mathrm{d}x-\arctan \left(x^{-n}y\frac{1}{\sqrt{a}}\right))$$

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6.2.24.13 [763] problem number 13

problem number 763

Added Feb. 4, 2019.

Problem 2.8.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$x{w}_x+(x^{2n}f(x)y^2+(a{x}^{n}f(x)-n)y+bf(x))w_y=0$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(-{\int }_1^x\frac{bf(K[5])\sqrt{\frac{K[5{]}^{2n}}{b}}}{K[5]}dK[5]-\frac{2\sqrt{b}{\tan }^{-1}\left(\frac{\sqrt{b}(\sqrt{\frac{a^2}{b}}-2y\sqrt{\frac{x^{2n}}{b}})}{\sqrt{4b-a^2}}\right)}{\sqrt{4b-a^2}})\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\_}\mathit{F}\mathit{1}(-2\frac{a}{\sqrt{a^2(a^2-4b)}}(a\mathrm{arctanh}\left(\frac{a(2y{x}^n+a)}{\sqrt{a^2(a^2-4b)}}\right)+1/2\sqrt{a^2(a^2-4b)}\int \frac{f\left(x\right)x^n}{x}\mathrm{d}x))$$

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