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 \(w(x,y)\)

\[ w_x + \left ( f(x) y+g(x) \right ) w_y = 0 \]

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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left ({\mathrm e}^{-\int f \left (x \right )d x} y -\int g \left (x \right ) {\mathrm e}^{-\int f \left (x \right )d x}d x \right )\]

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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 \(w(x,y)\)

\[ w_x + \left ( f(x) y+g(x) y^k \right ) w_y = 0 \]

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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\left (k -1\right ) \int g \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int f \left (x \right )d x}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int f \left (x \right )d x}\right )\]

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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 \(w(x,y)\)

\[ w_x + \left ( y^2+f(x) y -a^2 -a f(x)\right ) w_y = 0 \]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (a -y \right ) \int {\mathrm e}^{\int f \left (x \right )d x +2 a x}d x -{\mathrm e}^{\int f \left (x \right )d x +2 a x}}{a -y}\right )\]

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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 \(w(x,y)\)

\[ w_x + \left ( y^2+x f(x) y + f(x)\right ) w_y = 0 \]

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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y x \int {\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}d x +{\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}d x}{x y +1}\right )\]

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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 \(w(x,y)\)

\[ w_x - \left ( (k+1)x^k y^2-x^{k+1} f(x) y+f(x)\right ) w_y = 0 \]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-x^{k +1} {\mathrm e}^{\int \frac {x^{k +1} f \left (x \right ) x -2 k -2}{x}d x}+\int x^{-k -2} {\mathrm e}^{\int x^{k +1} f \left (x \right )d x}d x \left (y \,x^{k +1}-1\right ) \left (k +1\right )}{y \,x^{k +1}-1}\right )\]

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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 \(w(x,y)\)

\[ w_x + \left ( f(x) y^2+a y-a b- b^2 f(x)\right ) w_y = 0 \]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (b -y \right ) \int {\mathrm e}^{a x +2 b \int f \left (x \right )d x} f \left (x \right )d x -{\mathrm e}^{a x +2 b \int f \left (x \right )d x}}{b -y}\right )\]

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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 \(w(x,y)\)

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

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 \(w(x,y)\)

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

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 \(w(x,y)\)

\[ w_x + \left ( f(x) y^2+g(x) y-a^2 f(x)-a g(x)\right ) w_y = 0 \]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (a -y \right ) \int {\mathrm e}^{\int g \left (x \right )d x +2 a \int f \left (x \right )d x} f \left (x \right )d x -{\mathrm e}^{\int g \left (x \right )d x +2 a \int f \left (x \right )d x}}{a -y}\right )\]

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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 \(w(x,y)\)

\[ w_x + \left ( f(x) y^2+g(x) y+a n x^{n-1} - a x^n g(x)-a^2 x^{2 n} f(x)\right ) w_y = 0 \]

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 \(w(x,y)\)

\[ w_x + \left ( f(x) y^2-a x^n g(x) y+a n x^{n-1}+a^2 x^{2 n}(g(x)-f(x))\right ) w_y = 0 \]

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 \(w(x,y)\)

\[ x w_x + \left ( f(x) y^2+n y+a x^{2 n} f(x)\right ) w_y = 0 \]

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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\sqrt {a}\, \int f \left (x \right ) x^{n -1}d x -\arctan \left (\frac {x^{-n} y}{\sqrt {a}}\right )\right )\]

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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 \(w(x,y)\)

\[ x w_x + \left ( x^{2 n} f(x) y^2+(a x^n f(x)-n) y+b f(x)\right ) w_y = 0 \]

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]];
 

Failed

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {2 \left (a \,\operatorname {arctanh}\left (\frac {a \left (2 x^{n} y +a \right )}{\sqrt {a^{2} \left (a^{2}-4 b \right )}}\right )+\frac {\sqrt {a^{2} \left (a^{2}-4 b \right )}\, \int f \left (x \right ) x^{-1+n}d x}{2}\right ) a}{\sqrt {a^{2} \left (a^{2}-4 b \right )}}\right )\]

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