60.8.18 problem 1854

Internal problem ID [11853]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1854
Date solved : Monday, January 27, 2025 at 11:43:49 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }-f \left (y\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 51

dsolve(diff(y(x),x$2)-f(y(x))=0,y(x), singsol=all)
 

\begin{align*} \int _{}^{y}\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}\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: 0.028 (sec). Leaf size: 40

DSolve[-f[y[x]]+ D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}f(K[1])dK[1]}}dK[2]{}^2=(x+c_2){}^2,y(x)\right ] \]