Internal problem ID [8520]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 940.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class C], [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \ln \relax (x ) x +\ln \relax (x ) x^{2}-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \relax (x )-3 x^{2} \ln \relax (x )^{2} y+x^{3} \ln \relax (x )^{3}}{x \left (-y+x \ln \relax (x )-x \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 63
dsolve(diff(y(x),x) = 1/x*(y(x)*ln(x)*x+x^2*ln(x)-2*x*y(x)-x^2-y(x)^2-y(x)^3+3*x*y(x)^2*ln(x)-3*x^2*ln(x)^2*y(x)+x^3*ln(x)^3)/(-y(x)+x*ln(x)-x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x \left (\sqrt {c_{1}-2 x}\, \ln \relax (x )-\ln \relax (x )+1\right )}{\sqrt {c_{1}-2 x}-1} \\ y \relax (x ) = \frac {x \left (\sqrt {c_{1}-2 x}\, \ln \relax (x )+\ln \relax (x )-1\right )}{\sqrt {c_{1}-2 x}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.449 (sec). Leaf size: 57
DSolve[y'[x] == (-x^2 + x^2*Log[x] + x^3*Log[x]^3 - 2*x*y[x] + x*Log[x]*y[x] - 3*x^2*Log[x]^2*y[x] - y[x]^2 + 3*x*Log[x]*y[x]^2 - y[x]^3)/(x*(-x + x*Log[x] - y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (\log (x)-\frac {1}{1+\sqrt {-2 x+c_1}}\right ) \\ y(x)\to x \left (\log (x)+\frac {1}{-1+\sqrt {-2 x+c_1}}\right ) \\ y(x)\to x \log (x) \\ \end{align*}