7.2.29 8.6

7.2.29.1 [789] problem number 1

problem number 789

Added Feb. 7, 2019.

Problem 2.8.6.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

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

Mathematica ✗

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

sol=()

____________________________________________________________________________________

7.2.29.2 [790] problem number 2

problem number 790

Added Feb. 7, 2019.

Problem 2.8.6.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - (Derivative[1][f][x]*y^2 - f[x]*g[x]*y + 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)-( diff(f(x),x)*y^2 -f(x)*g(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'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{y(\int \frac{\left(\frac{d}{dx}f\left(x\right)\right){\mathrm{e}}^{\int f\left(x\right)g\left(x\right)dx}}{f{\left(x\right)}^2}dx)f\left(x\right)-{\mathrm{e}}^{-(\int \frac{-f{\left(x\right)}^{2}g\left(x\right)+2\frac{d}{dx}f\left(x\right)}{f\left(x\right)}dx)}f\left(x\right)-(\int \frac{\left(\frac{d}{dx}f\left(x\right)\right){\mathrm{e}}^{\int f\left(x\right)g\left(x\right)dx}}{f{\left(x\right)}^2}dx)}{yf\left(x\right)-1}\right)$$

____________________________________________________________________________________

7.2.29.3 [791] problem number 3

problem number 791

Added Feb. 7, 2019.

Problem 2.8.6.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

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

Mathematica ✓

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

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

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{(y-f\left(x\right))(\int g\left(x\right)dx)+1}{y-f\left(x\right)}\right)$$

____________________________________________________________________________________

7.2.29.4 [792] problem number 4

problem number 792

Added Feb. 7, 2019.

Problem 2.8.6.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(\frac{f^{\prime }(x)}{g(x)}{y}^2-\frac{g^{\prime }(x)}{f(x)})w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((Derivative[1][f][x]*y^2)/g[x] - Derivative[1][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)+(diff(f(x),x)/g(x)* y^2 - diff(g(x),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'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{(-yf{\left(x\right)}^2-f\left(x\right)g\left(x\right))(\int \frac{\frac{d}{dx}f\left(x\right)}{f{\left(x\right)}^{2}g\left(x\right)}dx)-1}{(yf\left(x\right)+g\left(x\right))f\left(x\right)}\right)$$

____________________________________________________________________________________

7.2.29.5 [793] problem number 5

problem number 793

Added Feb. 7, 2019.

Problem 2.8.6.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f[x]^2*D[w[x, y], x] + (Derivative[1][f][x]*y^2 - g[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 := f(x)^2*diff(w(x,y),x)+(diff(f(x),x)*y^2 -g(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'));
 

sol=()

____________________________________________________________________________________

7.2.29.6 [794] problem number 6

problem number 794

Added Feb. 7, 2019.

Problem 2.8.6.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(y^2-\frac{f^{″}(x)}{f(x)})w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - Derivative[2][f][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)+(y^2 - diff(f(x),x,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'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{(-yf{\left(x\right)}^2-\left(\frac{d}{dx}f\left(x\right)\right)f\left(x\right))(\int \frac{1}{f{\left(x\right)}^2}dx)-1}{(yf\left(x\right)+\frac{d}{dx}f\left(x\right))f\left(x\right)}\right)$$

____________________________________________________________________________________

7.2.29.7 [795] problem number 7

problem number 795

Added Feb. 7, 2019.

Problem 2.8.6.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$g(x)w_x+(af(x)g(x)y^3+(bf(x)g^3(x)+g^{\prime }(x))y+cf(x)g^4(x))w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  g[x]*D[w[x, y], x] + (a*f[x]*g[x]*y^3 + (b*f[x]*g[x]^3 + Derivative[1][g][x])*y + c*f[x]*g[x]^4)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  g(x)*diff(w(x,y),x)+(a*f(x)*g(x)*y^3 + (b*f(x)*g(x)^3 + diff(g(x),x))*y+ c*f(x)*g(x)^4)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}(\frac{b^3\ln \left(\frac{-\mathrm{RootOf}(a{c}^2{\mathit{\text{\_Z}}}^3+b^3\mathit{\text{\_Z}}-b^3)cg\left(x\right)-by}{cg\left(x\right)}\right)}{3\mathrm{RootOf}{(a{c}^2{\mathit{\text{\_Z}}}^3+b^3\mathit{\text{\_Z}}-b^3)}^{2}a{c}^2+b^3}-b(\int f\left(x\right)g{\left(x\right)}^{2}dx))$$

____________________________________________________________________________________

7.2.29.8 [796] problem number 8

problem number 796

Added Feb. 7, 2019.

Problem 2.8.6.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y^3+3f(x)h(x)y^2+(g(x)+3f(x)h^2(x))y+f(x)h^3(x)+g(x)h(x)-h^{\prime }(x))w_y=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^3 + 3*f[x]*h[x]*y^2 + (g[x] + 3*f[x]*h[x]^2)*y + f[x]*h[x]^3 + g[x]*h[x] - Derivative[1][h][x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+(f(x)*y^3+3*f(x)*h(x)*y^2+(g(x)+3*f(x)*h(x)^2)*y+ f(x)*h(x)^3 + g(x)* h(x) - diff(h(x),x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{2{(y+h\left(x\right))}^2(\int {\mathrm{e}}^{2(\int g\left(x\right)dx)}f\left(x\right)dx)+{\mathrm{e}}^{2(\int g\left(x\right)dx)}}{{(y+h\left(x\right))}^2}\right)$$

____________________________________________________________________________________

7.2.29.9 [797] problem number 9

problem number 797

Added Feb. 7, 2019.

Problem 2.8.6.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(\frac{g^{\prime }(x)}{f^2(x)(ag(x)+b{)}^3}{y}^3+\frac{f^{\prime }(x)}{f(x)}y+f(x)g^{\prime }(x))w_y=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + ((Derivative[1][g][x]*y^3)/(f[x]^2*(a*g[x] + b)^3) + (Derivative[1][f][x]*y)/f[x] + f[x]*Derivative[1][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)+(diff(g(x),x)/(f(x)^2 *(a*g(x)+b)^3)*y^3 + diff(f(x),x)/f(x) * y + f(x)*diff(g(x),x) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}(-\frac{a^3\ln \left(\frac{-(ag\left(x\right)+b)\mathrm{RootOf}(-a^3\mathit{\text{\_Z}}+{\mathit{\text{\_Z}}}^3+a^3)f\left(x\right)+ay}{(ag\left(x\right)+b)f\left(x\right)}\right)}{a^3-3\mathrm{RootOf}{(-a^3\mathit{\text{\_Z}}+{\mathit{\text{\_Z}}}^3+a^3)}^2}-\ln (ag\left(x\right)+b))$$

____________________________________________________________________________________

7.2.29.10 [798] problem number 10

problem number 798

Added Feb. 7, 2019.

Problem 2.8.6.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+((y-f(x))(y-g(x))(y-\frac{af(x)+bg(x)}{a+b})h(x)+\frac{y-g(x)}{f(x)-g(x)}{f}^{\prime }(x)+\frac{y-f(x)}{g(x)-f(x)}{g}^{\prime }(x))w_y=0$$

Mathematica ✗

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{((a+b)b\ln \left(\frac{9(a+b)(a^2+ab+b^2)(y-g\left(x\right))}{(f\left(x\right)-g\left(x\right))(2a+b)}\right)+(2(-\frac{(\int f{\left(x\right)}^{2}h\left(x\right)dx)}{2}-\frac{(\int g{\left(x\right)}^{2}h\left(x\right)dx)}{2}+\int f\left(x\right)g\left(x\right)h\left(x\right)dx)b+(a+b)\ln \left(\frac{9(a+b)(a^2+ab+b^2)(y-f\left(x\right))}{(f\left(x\right)-g\left(x\right))(a+2b)}\right))a-{(a+b)}^2\ln \left(\frac{9(a^2+ab+b^2)(-af\left(x\right)-bg\left(x\right)+(a+b)y)}{(f\left(x\right)-g\left(x\right))(a-b)}\right))(a^2+ab+b^2)}{3{(a+b)}^{2}ab}\right)$$

____________________________________________________________________________________

7.2.29.11 [799] problem number 11

problem number 799

Added Feb. 7, 2019.

Problem 2.8.6.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f(x)y^2+g^{\prime }(x)y+af(x)e^{2g(x)})w_y=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (f[x]*y^2 + Derivative[1][g][x]*y + a*f[x]*Exp[2*g[x]])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> a > 0], 60*10]];
 

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

Maple ✓

restart; 
pde := diff(w(x,y),x)+(f(x)*y^2 + diff(g(x),x)*y+ a*f(x)*exp(2*g(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'));
 

$$w(x,y)=\mathit{\text{\_F1}}(\sqrt{a}(\int {\mathrm{e}}^{g\left(x\right)}f\left(x\right)dx)-\arctan \left(\frac{y{\mathrm{e}}^{-g\left(x\right)}}{\sqrt{a}}\right))$$

____________________________________________________________________________________

7.2.29.12 [800] problem number 12

problem number 800

Added Feb. 7, 2019.

Problem 2.8.6.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$

$$w_x+(f^{\prime }(x)y^2+a{e}^{\lambda x}f(x)y+a{e}^{\lambda x})w_y=0$$

Mathematica ✗

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{-y(\int \frac{\left(\frac{d}{dx}f\left(x\right)\right){\mathrm{e}}^{a(\int {\mathrm{e}}^{\lambda x}f\left(x\right)dx)}}{f{\left(x\right)}^2}dx)f\left(x\right)-{\mathrm{e}}^{-(\int \frac{-a{\mathrm{e}}^{\lambda x}f{\left(x\right)}^2+2\frac{d}{dx}f\left(x\right)}{f\left(x\right)}dx)}f\left(x\right)-(\int \frac{\left(\frac{d}{dx}f\left(x\right)\right){\mathrm{e}}^{a(\int {\mathrm{e}}^{\lambda x}f\left(x\right)dx)}}{f{\left(x\right)}^2}dx)}{yf\left(x\right)+1}\right)$$

____________________________________________________________________________________