60.7.62 problem 1653 (book 6.62)

Internal problem ID [11651]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1653 (book 6.62)
Date solved : Tuesday, January 28, 2025 at 06:07:18 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=a \sqrt {{y^{\prime }}^{2}+b y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.247 (sec). Leaf size: 41

dsolve(diff(diff(y(x),x),x)=a*sqrt(diff(y(x),x)^2+b*y(x)^2),y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (x -\int _{}^{\textit {\_Z}}-\frac {1}{\textit {\_f}^{2}-a \sqrt {\textit {\_f}^{2}+b}}d \textit {\_f} +c_{1} \right )d x +c_{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.348 (sec). Leaf size: 76

DSolve[D[y[x],{x,2}]==a*Sqrt[D[y[x],x]^2+b*y[x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int \frac {\text {$\#$1}}{K[1] \left (\frac {\text {$\#$1}^2}{K[1]^2}-a \sqrt {\frac {\text {$\#$1}^2}{K[1]^2}+b}\right )}d\frac {\text {$\#$1}}{K[1]}\&\right ][c_1-\log (K[1])]}dK[1]=x-c_2,y(x)\right ] \]