Internal problem ID [10320]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 61.
Transformation of variables. Page 143
Problem number: Ex 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y y^{\prime \prime }-\left (y^{\prime }\right )^{2}-y^{2} \ln \relax (y)+x^{2} y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 31
dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2=y(x)^2*ln(y(x))-x^2*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{x^{2}} {\mathrm e}^{-\frac {{\mathrm e}^{-x} {\mathrm e}^{2 x} c_{1}}{2}} {\mathrm e}^{2} {\mathrm e}^{\frac {c_{2} {\mathrm e}^{-x}}{2}} \]
✓ Solution by Mathematica
Time used: 0.111 (sec). Leaf size: 30
DSolve[y[x]*y''[x]-y'[x]^2==y[x]^2*Log[y[x]]-x^2*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{x^2-\frac {c_1 e^x}{2}-c_2 e^{-x}+2} \\ \end{align*}