81.2.3 problem 3

Internal problem ID [18611]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 12:04:02 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Solution by Maple

Time used: 0.043 (sec). Leaf size: 91

dsolve((1+diff(y(x),x)^2)^(3/2)=r*diff(y(x),x$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 (r +x +c_{1} \right ) \left (-r +x +c_{1} \right )}{\sqrt {-c_{1}^{2}-2 c_{1} x +r^{2}-x^{2}}}+c_{2} \\ y \left (x \right ) &= \frac {\left (r +x +c_{1} \right ) \left (r -x -c_{1} \right )}{\sqrt {-c_{1}^{2}-2 c_{1} x +r^{2}-x^{2}}}+c_{2} \\ \end{align*}

Solution by Mathematica

Time used: 0.641 (sec). Leaf size: 71

DSolve[(1+D[y[x],x]^2)^(3/2)==r*D[y[x],{x,2}],y[x],x,IncludeSingularSolutions -> True]
 
\begin{align*} y(x)\to c_2-i \sqrt {\left (-1+c_1{}^2\right ) r^2+2 c_1 r x+x^2} \\ y(x)\to i \sqrt {\left (-1+c_1{}^2\right ) r^2+2 c_1 r x+x^2}+c_2 \\ \end{align*}