81.1.5 problem 4

Internal problem ID [18600]
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 : 4
Date solved : Tuesday, January 28, 2025 at 12:03:40 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 1+{y^{\prime }}^{2}+\frac {m y^{\prime \prime }}{\sqrt {{y^{\prime }}^{2}+1}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 75

dsolve(1+diff(y(x),x)^2+m/sqrt(1+diff(y(x),x)^2)*diff(y(x),x$2)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.680 (sec). Leaf size: 71

DSolve[1+D[y[x],x]^2+m/Sqrt[1+D[y[x],x]^2]*D[y[x],{x,2}]==0,y[x],x,IncludeSingularSolutions -> True]
 

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