Internal problem ID [4178]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter 1, Nature and meaning of a differential equation between two variables. page
12
Problem number: 9.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+1\right )^{3}-a^{2} \left (y^{\prime \prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.07 (sec). Leaf size: 96
dsolve((diff(y(x),x)^2+1)^3=a^2*(diff(y(x),x$2))^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = -\frac {\left (a +x +c_{1}\right ) \left (a -x -c_{1}\right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ y \relax (x ) = \frac {\left (a +x +c_{1}\right ) \left (a -x -c_{1}\right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.391 (sec). Leaf size: 129
DSolve[(y'[x]^2+1)^3==a^2*(y''[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i \sqrt {(a+a (-c_1)+x) (x-a (1+c_1))} \\ y(x)\to i \sqrt {(a+a (-c_1)+x) (x-a (1+c_1))}+c_2 \\ y(x)\to c_2-i \sqrt {(x+a (-1+c_1)) (a+a c_1+x)} \\ y(x)\to i \sqrt {(x+a (-1+c_1)) (a+a c_1+x)}+c_2 \\ \end{align*}