6.2.14 5.2

6.2.14.1 [616] problem number 1

problem number 616

Added January 14, 2019.

Problem 2.5.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a x^n \ln ^k(\lambda y) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {(-\log (\lambda y))^k \log ^{-k}(\lambda y) \Gamma (1-k,-\log (\lambda y))}{\lambda }-\frac {a x^{n+1}}{n+1}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+a*x^n*ln(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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (-n -1\right ) \int \ln \left (\lambda y \right )^{-k}d y +a \,x^{n +1}}{a}\right )\]

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6.2.14.2 [617] problem number 2

problem number 617

Added January 14, 2019.

Problem 2.5.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a y^n \ln ^k(\lambda x) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a \log ^k(\lambda x) (-\log (\lambda x))^{-k} \Gamma (k+1,-\log (\lambda x))}{\lambda }-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (y^{1-n}+\left (n -1\right ) a \int \ln \left (\lambda x \right )^{k}d x \right )\]

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6.2.14.3 [618] problem number 3

problem number 618

Added January 14, 2019.

Problem 2.5.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✗

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

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

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6.2.14.4 [619] problem number 4

problem number 619

Added January 14, 2019.

Problem 2.5.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.14.5 [620] problem number 5

problem number 620

Added January 14, 2019.

Problem 2.5.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*x^n*y^2 - a*b*x^(n + 1)*y*Log[x] + b*Log[x] + b)*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*x^n*y^2- a*b*x^(n+1)*y*ln(x)  + b*ln(x) + b)*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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6.2.14.6 [621] problem number 6

problem number 621

Added January 14, 2019.

Problem 2.5.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.14.7 [622] problem number 7

problem number 622

Added January 14, 2019.

Problem 2.5.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Log[x]^n*y^2 + b*m*x^(m - 1) - a*b^2*x^(2*m)*Log[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)+(a *ln(x)^n*y^2 + b*m*x^(m-1) - a*b^2*x^(2*m)* ln(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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6.2.14.8 [623] problem number 8

problem number 623

Added January 14, 2019.

Problem 2.5.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Log[x]^n*y^2 - a*b*x*y*Log[x]^(n + 1) + b*Log[x] + b)*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*ln(x)^n*y^2 - a*b*x*y*(ln(x))^(n+1) + b*ln(x)+ b )*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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6.2.14.9 [624] problem number 9

problem number 624

Added January 14, 2019.

Problem 2.5.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (a (\ln x)^k (y - b x^n-c)^3 + b n x^{n-1} \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {(-\log (x))^{-k} \left (2 a \log ^k(x) \Gamma (k+1,-\log (x)) \left (b x^n+c-y\right )^2+(-\log (x))^k\right )}{\left (b x^n+c-y\right )^2}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x,y),x)+(a*(ln(x))^k*(y - b*x^n-c)^3 + b*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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {1+2 \left (x^{2 n} b^{2}+2 \left (b \,x^{n}+\frac {c}{2}-\frac {y}{2}\right ) \left (c -y \right )\right ) a \int \ln \left (x \right )^{k}d x}{\left (b \,x^{n}+c -y \right )^{2}}\right )\]

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6.2.14.10 [625] problem number 10

problem number 625

Added January 14, 2019.

Problem 2.5.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (a (\ln x)^n y^2 + b(\ln x)^m y+ b c (\ln x)^m - a c^2 (\ln x)^n \right ) w_y = 0 \]

Mathematica ✗

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

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

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6.2.14.11 [626] problem number 11

problem number 626

Added January 14, 2019.

Problem 2.5.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {a y+b \log (x)}{a \sqrt {\frac {b}{a^3}}}\right )-a^2 \sqrt {\frac {b}{a^3}} \log (x)\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-\ln \left (x \right ) \sqrt {a b}+\arctan \left (\frac {a \left (a y +b \ln \left (x \right )\right )}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\]

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6.2.14.12 [627] problem number 12

problem number 627

Added January 14, 2019.

Problem 2.5.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left (x y^2 - A^2 x (\ln \beta x)^2 + A \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (x*y^2 - A^2*x*Log[beta*x]^2 + A)*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*y^2 - A^2*x*(ln(beta*x))^2 + A) *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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6.2.14.13 [628] problem number 13

problem number 628

Added January 14, 2019.

Problem 2.5.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (x*y^2 - A^2*x*Log[beta*x]^(2*k) + k*A*Log[beta*x]^(k - 1))*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*y^2 - A^2*x*(ln(beta*x))^(2*k) + k*A*(ln(beta*x))^(k-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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6.2.14.14 [629] problem number 14

problem number 629

Added January 14, 2019.

Problem 2.5.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (a*x^n*y^2 + b - a*b^2*x^n*Log[x]^2)*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)+(a*x^n*y^2 + b - a*b^2*x^n*(ln(x))^2)*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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6.2.14.15 [630] problem number 15

problem number 630

Added January 14, 2019.

Problem 2.5.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a (\ln (\lambda x))^m y^2 + k y+ a b^2 x^{2 k} (\ln (\lambda x))^m \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^{m+1}(\lambda x) (-k \log (\lambda x))^{-m-1} \Gamma (m+1,-k \log (\lambda x))+\arctan \left (\frac {y x^{-k}}{\sqrt {b^2}}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+(a*(ln(lambda*x))^m*y^2 + k*y+ a*b^2*x^(2*k)* (ln(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'));
 

\[w \left (x , y\right ) = f_{1} \left (b a \int \ln \left (\lambda x \right )^{m} x^{k -1}d x -\arctan \left (\frac {x^{-k} y}{b}\right )\right )\]

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6.2.14.16 [631] problem number 16

problem number 631

Added January 14, 2019.

Problem 2.5.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a b x^n \log (x)+a y x^n+n}{b n \log (x)+n y}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {a \left (y +b \ln \left (x \right )\right ) x^{n}+n}{n \left (y +b \ln \left (x \right )\right )}\right )\]

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6.2.14.17 [632] problem number 17

problem number 632

Added January 14, 2019.

Problem 2.5.2.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\left (\sqrt {b^2-4 a c}+2 a y x^n+b\right ) e^{\frac {x^n \sqrt {b^2-4 a c} (n \log (x)-1)}{n^2}}}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {2 \left (-\frac {\sqrt {4 b^{2} a c -b^{4}}\, x^{n} \left (\ln \left (x \right ) n -1\right )}{2}+b \,n^{2} \arctan \left (\frac {b^{2}+2 x^{n} a y b}{\sqrt {4 b^{2} a c -b^{4}}}\right )\right ) b}{\sqrt {4 b^{2} a c -b^{4}}\, n^{2}}\right )\]

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6.2.14.18 [633] problem number 18

problem number 633

Added January 14, 2019.

Problem 2.5.2.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^k w_x + \left (a y^n (\ln x)^m + b y (\ln x)^s \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left ((n-1) \int _1^xa \exp \left (-b (k-1)^{-s-1} (n-1) \Gamma (s+1,(k-1) \log (K[1]))\right ) K[1]^{-k} \log ^m(K[1])dK[1]+y^{1-n} \exp \left (-b (n-1) (k-1)^{-s-1} \Gamma (s+1,(k-1) \log (x))\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int x^{-k} \ln \left (x \right )^{m} {\mathrm e}^{b \int x^{-k} \ln \left (x \right )^{s}d x \left (n -1\right )}d x +y^{1-n} {\mathrm e}^{b \int x^{-k} \ln \left (x \right )^{s}d x \left (n -1\right )}\right )\]

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6.2.14.19 [634] problem number 19

problem number 634

Added January 14, 2019.

Problem 2.5.2.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a \ln x+b) w_x + \left (y^2+ c(\ln x)^n y- \lambda ^2 + \lambda c( \ln x)^n \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

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

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

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6.2.14.20 [635] problem number 20

problem number 635

Added January 14, 2019.

Problem 2.5.2.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a \ln x+b) w_x + \left ((\ln x)^n y^2- c y - \lambda ^2 ( \ln x)^n + c \lambda \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

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

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

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6.2.14.21 [636] problem number 21

problem number 636

Added January 14, 2019.

Problem 2.5.2.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^2 \ln (a x) w_x - \left ( x^2 y^2 \ln (a x) + 1\right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a (x y \log (a x)-1)}{\operatorname {LogIntegral}(a x) (1-x y \log (a x))+a x^2 y}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {-\ln \left (a x \right ) x y +1}{y \,\operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right ) \ln \left (a x \right ) x +y a \,x^{2}-\operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right )}\right )\]

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6.2.14.22 [637] problem number 22

problem number 637

Added January 14, 2019.

Problem 2.5.2.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \ln ^k(\lambda x) w_x + \left ( a y^n + b y \ln ^m x\right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left ((n-1) \int _1^xa \exp \left ((n-1) \int _1^{K[2]}b \log ^m(K[1]) (\log (\lambda )+\log (K[1]))^{-k}dK[1]\right ) (\log (\lambda )+\log (K[2]))^{-k}dK[2]+y^{1-n} \exp \left ((n-1) \int _1^xb \log ^m(K[1]) (\log (\lambda )+\log (K[1]))^{-k}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int \ln \left (\lambda x \right )^{-k} {\mathrm e}^{b \int \ln \left (\lambda x \right )^{-k} \ln \left (x \right )^{m}d x \left (n -1\right )}d x +y^{1-n} {\mathrm e}^{b \int \ln \left (\lambda x \right )^{-k} \ln \left (x \right )^{m}d x \left (n -1\right )}\right )\]

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6.2.14.23 [638] problem number 23

problem number 638

Added January 14, 2019.

Problem 2.5.2.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \ln ^k(\lambda x) w_x + \left ( a y^n \ln ^m x + b y \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left ((n-1) \int _1^xa \exp \left (\frac {b (n-1) \Gamma (1-k,-\log (\lambda )-\log (K[1])) (-\log (\lambda )-\log (K[1]))^k (\log (\lambda )+\log (K[1]))^{-k}}{\lambda }\right ) \log ^m(K[1]) (\log (\lambda )+\log (K[1]))^{-k}dK[1]+y^{1-n} \exp \left (\frac {b (n-1) (-\log (\lambda )-\log (x))^k (\log (\lambda )+\log (x))^{-k} \Gamma (1-k,-\log (\lambda )-\log (x))}{\lambda }\right )\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int \ln \left (x \right )^{m} \ln \left (\lambda x \right )^{-k} {\mathrm e}^{b \int \ln \left (\lambda x \right )^{-k}d x \left (n -1\right )}d x +y^{1-n} {\mathrm e}^{b \int \ln \left (\lambda x \right )^{-k}d x \left (n -1\right )}\right )\]

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