Internal problem ID [10319]
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 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }-\left (y^{\prime } x -y\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 19
dsolve(x^3*diff(y(x),x$2)-(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\[ y \relax (x ) = -x \ln \left (\frac {x c_{1}-c_{2}}{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 21
DSolve[x^3*y''[x]-(x*y'[x]-y[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \log \left (-\frac {c_2 x+c_1}{x}\right ) \\ \end{align*}