Internal problem ID [9331]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1752.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime } y-5 \left (y^{\prime }\right )^{2}+a y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.041 (sec). Leaf size: 37
dsolve(4*diff(diff(y(x),x),x)*y(x)-5*diff(y(x),x)^2+a*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {16 \,{\mathrm e}^{x \sqrt {a}} a^{2}}{\left ({\mathrm e}^{\frac {x \sqrt {a}}{2}} c_{1}-c_{2}\right )^{4}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.129 (sec). Leaf size: 26
DSolve[a*y[x]^2 - 5*y'[x]^2 + 4*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \text {sech}^4\left (\frac {1}{4} \sqrt {a} (x-4 c_1)\right ) \\ \end{align*}