75.13.8 problem 325

Internal problem ID [16913]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 13. Basic concepts and definitions. Exercises page 98
Problem number : 325
Date solved : Tuesday, January 28, 2025 at 09:40:44 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Solution by Maple

Time used: 0.168 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.239 (sec). Leaf size: 59

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

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