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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {(-\log (\lambda y))^k \log ^{-k}(\lambda y) \operatorname {Gamma}(1-k,-\log (\lambda y))}{\lambda }-\frac {a x^{n+1}}{n+1}\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a \,x^{n +1}+\left (-n -1\right ) \left (\int \ln \left (\lambda y \right )^{-k}d y \right )}{a}\right )\]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {a \log ^k(\lambda x) (-\log (\lambda x))^{-k} \operatorname {Gamma}(k+1,-\log (\lambda x))}{\lambda }-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\left (\left (n -1\right ) a \,y^{n} \left (\int \ln \left (\lambda x \right )^{k}d x \right )+y \right ) y^{-n}\right )\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\left (\beta x \right )^{a x} {\mathrm e}^{-\left (a -2 b \right ) x}+\left (b -y \right ) \left (\int \left (\beta x \right )^{a x} {\mathrm e}^{-\left (a -2 b \right ) x}d x \right )}{b -y}\right )\]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\int _1^x\frac {\exp \left (\frac {2^{-m-1} a \operatorname {Gamma}(m+1,-2 (\log (b)+\log (K[1]))) (-\log (b)-\log (K[1]))^{-m} (\log (b)+\log (K[1]))^m}{b^2}\right )}{K[1]^2}dK[1]-\frac {\exp \left (\frac {a 2^{-m-1} (-\log (b)-\log (x))^{-m} (\log (b)+\log (x))^m \operatorname {Gamma}(m+1,-2 (\log (b)+\log (x)))}{b^2}\right )}{x (x y+1)}\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {x y \left (\int {\mathrm e}^{\int \frac {a \,x^{2} \ln \left (b x \right )^{m}-2}{x}d x}d x \right )+x \,{\mathrm e}^{\int \frac {a \,x^{2} \ln \left (b x \right )^{m}-2}{x}d x}+\int {\mathrm e}^{\int \frac {a \,x^{2} \ln \left (b x \right )^{m}-2}{x}d x}d x}{x y +1}\right )\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-x^{n +1} {\mathrm e}^{\int \frac {a x \,x^{n +1} \ln \left (x \right )^{m}-2 n -2}{x}d x}+\left (y \,x^{n +1}-1\right ) \left (n +1\right ) \left (\int \frac {x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 n \ln \left (x \right )}}{x^{2}}d x \right )}{y \,x^{n +1}-1}\right )\]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \left (b x^n+c-y\right )^2 \int _1^xa \log ^k(K[1])dK[1]+1}{\left (b x^n+c-y\right )^2}\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {2 \left (b^{2} x^{2 n}+2 \left (c -y \right ) \left (b \,x^{n}+\frac {c}{2}-\frac {y}{2}\right )\right ) a \left (\int \ln \left (x \right )^{k}d x \right )+1}{\left (b \,x^{n}+c -y \right )^{2}}\right )\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\left (c +y \right ) a \left (\int \ln \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 a c \ln \left (x \right )^{n}-b \ln \left (x \right )^{m}\right )d x \right )}d x \right )-{\mathrm e}^{-\left (\int \left (2 a c \ln \left (x \right )^{n}-b \ln \left (x \right )^{m}\right )d x \right )}}{c +y}\right )\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\arctan \left (\frac {\left (a y +b \ln \left (x \right )\right ) a}{\sqrt {a b}}\right )-\sqrt {a b}\, \ln \left (x \right )}{\sqrt {a b}\, a}\right )\]
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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=()
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left (a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^{m+1}(\lambda x) (-k \log (\lambda x))^{-m-1} \operatorname {Gamma}(m+1,-k \log (\lambda x))+\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (a b \left (\int x^{k -1} \ln \left (\lambda x \right )^{m}d x \right )-\arctan \left (\frac {y \,x^{-k}}{b}\right )\right )\]
____________________________________________________________________________________
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (b \ln \left (x \right )+y \right ) a \,x^{n}+n}{\left (b \ln \left (x \right )+y \right ) n}\right )\]
____________________________________________________________________________________
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {\left (-2 b \,n^{2} \arctan \left (\frac {2 a b y \,x^{n}+b^{2}}{\sqrt {4 a c \,b^{2}-b^{4}}}\right )+\sqrt {4 a c \,b^{2}-b^{4}}\, \left (n \ln \left (x \right )-1\right ) x^{n}\right ) b}{\sqrt {4 a c \,b^{2}-b^{4}}\, n^{2}}\right )\]
____________________________________________________________________________________
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]];
\[\left \{\left \{w(x,y)\to c_1\left ((n-1) \int _1^xa \exp \left (-b (k-1)^{-s-1} (n-1) \operatorname {Gamma}(s+1,(k-1) \log (K[1]))\right ) K[1]^{-k} \log ^m(K[1])dK[1]+y^{1-n} \exp \left (-b (n-1) (k-1)^{-s-1} \operatorname {Gamma}(s+1,(k-1) \log (x))\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\left (n -1\right ) a \left (\int x^{-k} \ln \left (x \right )^{m} {\mathrm e}^{\left (n -1\right ) b \left (\int x^{-k} \ln \left (x \right )^{s}d x \right )}d x \right )+y^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int x^{-k} \ln \left (x \right )^{s}d x \right )}\right )\]
____________________________________________________________________________________
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\lambda \left (\int \frac {{\mathrm e}^{\int \frac {c \ln \left (x \right )^{n}-2 \lambda }{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x \right ) {\mathrm e}^{\int \frac {-c \ln \left (x \right )^{n}+2 \lambda }{a \ln \left (x \right )+b}d x +\int \frac {c \ln \left (x \right )^{n}-2 \lambda }{a \ln \left (x \right )+b}d x}-y \left (\int \frac {{\mathrm e}^{\int \frac {c \ln \left (x \right )^{n}-2 \lambda }{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x \right )-{\mathrm e}^{\int \frac {c \ln \left (x \right )^{n}-2 \lambda }{a \ln \left (x \right )+b}d x}}{\lambda \,{\mathrm e}^{\int \frac {-c \ln \left (x \right )^{n}+2 \lambda }{a \ln \left (x \right )+b}d x +\int \frac {c \ln \left (x \right )^{n}-2 \lambda }{a \ln \left (x \right )+b}d x}+y}\right )\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\lambda \left (\int \frac {\ln \left (x \right )^{n} {\mathrm e}^{\int \frac {2 \lambda \ln \left (x \right )^{n}-c}{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x \right ) {\mathrm e}^{\int \frac {-2 \lambda \ln \left (x \right )^{n}+c}{a \ln \left (x \right )+b}d x +\int \frac {2 \lambda \ln \left (x \right )^{n}-c}{a \ln \left (x \right )+b}d x}+y \left (\int \frac {\ln \left (x \right )^{n} {\mathrm e}^{\int \frac {2 \lambda \ln \left (x \right )^{n}-c}{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x \right )+{\mathrm e}^{\int \frac {2 \lambda \ln \left (x \right )^{n}-c}{a \ln \left (x \right )+b}d x}}{\lambda \,{\mathrm e}^{\int \frac {-2 \lambda \ln \left (x \right )^{n}+c}{a \ln \left (x \right )+b}d x +\int \frac {2 \lambda \ln \left (x \right )^{n}-c}{a \ln \left (x \right )+b}d x}-y}\right )\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {x y \ln \left (a x \right )-1}{a \,x^{2} y +x y \Ei \left (1, -\ln \left (a x \right )\right ) \ln \left (a x \right )-\Ei \left (1, -\ln \left (a x \right )\right )}\right )\]
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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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\left (n -1\right ) a \left (\int \ln \left (\lambda x \right )^{-k} {\mathrm e}^{\left (n -1\right ) b \left (\int \ln \left (x \right )^{m} \ln \left (\lambda x \right )^{-k}d x \right )}d x \right )+y^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \ln \left (x \right )^{m} \ln \left (\lambda x \right )^{-k}d x \right )}\right )\]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left ((n-1) \int _1^xa \exp \left (\frac {b (n-1) \operatorname {Gamma}(1-k,-\log (\lambda )-\log (K[1])) (-\log (\lambda )-\log (K[1]))^k (\log (\lambda )+\log (K[1]))^{-k}}{\lambda }\right ) \log ^m(K[1]) (\log (\lambda )+\log (K[1]))^{-k}dK[1]+y^{1-n} \exp \left (\frac {b (n-1) (-\log (\lambda )-\log (x))^k (\log (\lambda )+\log (x))^{-k} \operatorname {Gamma}(1-k,-\log (\lambda )-\log (x))}{\lambda }\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\left (n -1\right ) a \left (\int \ln \left (x \right )^{m} \ln \left (\lambda x \right )^{-k} {\mathrm e}^{\left (n -1\right ) b \left (\int \ln \left (\lambda x \right )^{-k}d x \right )}d x \right )+y^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \ln \left (\lambda x \right )^{-k}d x \right )}\right )\]
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