60.7.63 problem 1654 (book 6.63)

Internal problem ID [11652]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1654 (book 6.63)
Date solved : Monday, January 27, 2025 at 11:29:18 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=a \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}} \end{align*}

✓ Solution by Maple

Time used: 0.200 (sec). Leaf size: 57

dsolve(diff(diff(y(x),x),x)=a*(diff(y(x),x)^2+1)^(3/2),y(x), singsol=all)
 

\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= \frac {\left (-1+\left (x +c_{1} \right )^{2} a^{2}\right ) \sqrt {-\frac {1}{-1+\left (x +c_{1} \right )^{2} a^{2}}}+c_{2} a}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.571 (sec). Leaf size: 75

DSolve[-(a*(1 + D[y[x],x]^2)^(3/2)) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_2-\frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a} \\ y(x)\to \frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a}+c_2 \\ \end{align*}