Internal problem ID [9275]
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)]`]]
\[ \boxed {y^{\prime }-\frac {\ln \left (x \right ) y x +x^{2} \ln \left (x \right )-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 y \ln \left (x \right )^{2} x^{2}+\ln \left (x \right )^{3} x^{3}}{x \left (-y+x \ln \left (x \right )-x \right )}=0} \]
✓ Solution by Maple
Time used: 0.0 (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 \left (x \right ) = \frac {x \left (\sqrt {-2 x +c_{1}}\, \ln \left (x \right )-\ln \left (x \right )+1\right )}{\sqrt {-2 x +c_{1}}-1} y \left (x \right ) = \frac {x \left (\sqrt {-2 x +c_{1}}\, \ln \left (x \right )+\ln \left (x \right )-1\right )}{\sqrt {-2 x +c_{1}}+1} \end{align*}
✓ Solution by Mathematica
Time used: 0.45 (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*}