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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {(-\log (\lambda y))^k \log ^{-k}(\lambda y) \text {Gamma}(1-k,-\log (\lambda y))}{\lambda }-\frac {a x^{n+1}}{n+1}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( -n-1 \right ) \int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y+a{x}^{n+1}}{a}} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a \log ^k(\lambda x) (-\log (\lambda x))^{-k} \text {Gamma}(k+1,-\log (\lambda x))}{\lambda }-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {a{y}^{n} \left ( n-1 \right ) \int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+y}{{y}^{n}}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( b-y \right ) \int \! \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}\,{\rm d}x- \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}}{b-y}} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\int _1^x\frac {\exp \left (\frac {2^{-m-1} a \text {Gamma}(m+1,-2 (\log (b)+\log (K[1]))) (-\log (b)-\log (K[1]))^{-m} (\log (b)+\log (K[1]))^m}{b^2}\right )}{K[1]^2}dK[1]-\frac {\exp \left (\frac {a 2^{-m-1} (-\log (b)-\log (x))^{-m} (\log (b)+\log (x))^m \text {Gamma}(m+1,-2 (\log (b)+\log (x)))}{b^2}\right )}{x (x y+1)}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{xy+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac {a \left ( \ln \left ( xb \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {a \left ( \ln \left ( xb \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac {a \left ( \ln \left ( xb \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]
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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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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]];
Failed
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{y{x}^{n+1}-1} \left ( \left ( y{x}^{n+1}-1 \right ) \left ( n+1 \right ) \int \!{\frac {{x}^{n}{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}}{{x}^{2}}}\,{\rm d}x-{{\rm e}^{\int \!{\frac {a{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}x-2\,n-2}{x}}\,{\rm d}x}}{x}^{n+1} \right ) } \right ) \]
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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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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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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {(-\log (x))^{-k} \left (2 a \log ^k(x) \text {Gamma}(k+1,-\log (x)) \left (b x^n+c-y\right )^2+(-\log (x))^k\right )}{\left (b x^n+c-y\right )^2}\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1+2\, \left ( {b}^{2}{x}^{2\,n}+2\, \left ( c-y \right ) \left ( b{x}^{n}+c/2-y/2 \right ) \right ) a\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x}{ \left ( b{x}^{n}+c-y \right ) ^{2}}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-a \left ( c+y \right ) \int \! \left ( \ln \left ( x \right ) \right ) ^{n}{{\rm e}^{-\int \!2\, \left ( \ln \left ( x \right ) \right ) ^{n}ac- \left ( \ln \left ( x \right ) \right ) ^{m}b\,{\rm d}x}}\,{\rm d}x-{{\rm e}^{-\int \!2\, \left ( \ln \left ( x \right ) \right ) ^{n}ac- \left ( \ln \left ( x \right ) \right ) ^{m}b\,{\rm d}x}}}{c+y}} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {a y+b \log (x)}{a \sqrt {\frac {b}{a^3}}}\right )-a^2 \sqrt {\frac {b}{a^3}} \log (x)\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a\sqrt {ab}} \left ( -\ln \left ( x \right ) \sqrt {ab}+\arctan \left ( {\frac {a \left ( ya+b\ln \left ( x \right ) \right ) }{\sqrt {ab}}} \right ) \right ) } \right ) \]
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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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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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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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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]];
\[\left \{\left \{w(x,y)\to c_1\left (a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^{m+1}(\lambda x) (-k \log (\lambda x))^{-m-1} \text {Gamma}(m+1,-k \log (\lambda x))+\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( ab\int \!{x}^{k-1} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{m}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {a \left ( y+b\ln \left ( x \right ) \right ) {x}^{n}+n}{n \left ( y+b\ln \left ( x \right ) \right ) }} \right ) \]
____________________________________________________________________________________
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {b}{\sqrt {4\,ac{b}^{2}-{b}^{4}}{n}^{2}} \left ( -2\,b{n}^{2}\arctan \left ( {\frac {2\,bay{x}^{n}+{b}^{2}}{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) +\sqrt {4\,ac{b}^{2}-{b}^{4}}{x}^{n} \left ( n\ln \left ( x \right ) -1 \right ) \right ) } \right ) \]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left ((n-1) \int _1^xa \exp \left (-b (k-1)^{-s-1} (n-1) \text {Gamma}(s+1,(k-1) \log (K[1]))\right ) K[1]^{-k} \log ^m(K[1])dK[1]+y^{1-n} \exp \left (-b (n-1) (k-1)^{-s-1} \text {Gamma}(s+1,(k-1) \log (x))\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{{\rm e}^{b\int \!{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{s}\,{\rm d}x \left ( n-1 \right ) }}{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x+{y}^{-n+1}{{\rm e}^{b\int \!{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{s}\,{\rm d}x \left ( n-1 \right ) }} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( -\int \!{\frac {1}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!{\frac {- \left ( \ln \left ( x \right ) \right ) ^{n}c+2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}-y\int \!{\frac {1}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x-{{\rm e}^{\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}} \right ) \left ( {{\rm e}^{\int \!{\frac {- \left ( \ln \left ( x \right ) \right ) ^{n}c+2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}c-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}\lambda +y \right ) ^{-1}} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( -\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!{\frac {-2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda +c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}+y\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}} \right ) \left ( {{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!{\frac {-2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda +c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}\lambda -y \right ) ^{-1}} \right ) \]
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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]];
Failed
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {xy\ln \left ( ax \right ) -1}{y\ln \left ( ax \right ) \Ei \left ( 1,-\ln \left ( ax \right ) \right ) x+a{x}^{2}y-\Ei \left ( 1,-\ln \left ( ax \right ) \right ) }} \right ) \]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{{\rm e}^{b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x \left ( n-1 \right ) }} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x+{y}^{-n+1}{{\rm e}^{b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x \left ( n-1 \right ) }} \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left ((n-1) \int _1^xa \exp \left (\frac {b (n-1) \text {Gamma}(1-k,-\log (\lambda )-\log (K[1])) (-\log (\lambda )-\log (K[1]))^k (\log (\lambda )+\log (K[1]))^{-k}}{\lambda }\right ) \log ^m(K[1]) (\log (\lambda )+\log (K[1]))^{-k}dK[1]+y^{1-n} \exp \left (\frac {b (n-1) (-\log (\lambda )-\log (x))^k (\log (\lambda )+\log (x))^{-k} \text {Gamma}(1-k,-\log (\lambda )-\log (x))}{\lambda }\right )\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( a \left ( n-1 \right ) \int \!{{\rm e}^{b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x+{y}^{-n+1}{{\rm e}^{b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }} \right ) \]
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