Internal problem ID [10299]
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 58.
Independent variable absent. Page 135
Problem number: Ex 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}-y^{\prime } y^{2}=0} \]
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 32
dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2-y(x)^2*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 \\ y \left (x \right ) = -\frac {c_{1} {\mathrm e}^{c_{2} c_{1}} {\mathrm e}^{x c_{1}}}{-1+{\mathrm e}^{c_{2} c_{1}} {\mathrm e}^{x c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.123 (sec). Leaf size: 25
DSolve[y[x]*y''[x]-y'[x]^2-y[x]^2*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \left (-1+\frac {1}{1-e^{c_1 (x+c_2)}}\right ) \\ \end{align*}