Internal problem ID [9433]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1854.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-f \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 51
dsolve(diff(y(x),x$2)-f(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{\sqrt {2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )+c_{1}}}d \textit {\_b} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {1}{\sqrt {2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )+c_{1}}}d \textit {\_b} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 40
DSolve[-f[y[x]]+ y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}f(K[1])dK[1]}}dK[2]{}^2=(x+c_2){}^2,y(x)\right ] \]