81.1.2 problem 1 (b)

Internal problem ID [18597]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter I. Introduction. Exercises at page 13
Problem number : 1 (b)
Date solved : Tuesday, January 28, 2025 at 12:03:34 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Solution by Maple

Time used: 0.083 (sec). Leaf size: 57

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

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

✓ Solution by Mathematica

Time used: 0.553 (sec). Leaf size: 75

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

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