Internal problem ID [11313]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first.
Article 57. Dependent variable absent. Page 132
Problem number: Ex 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 94
dsolve((x*diff(y(x),x$3)-diff(y(x),x$2))^2=diff(y(x),x$3)^2+1,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{6}+c_{1} x +\frac {x \arcsin \left (x \right )}{2}+c_{2} \\ y \left (x \right ) &= -\frac {x^{2} \sqrt {-x^{2}+1}}{6}-\frac {\sqrt {-x^{2}+1}}{3}-\frac {x \arcsin \left (x \right )}{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\sqrt {c_{1}^{2}-1}\, x^{3}}{6}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.241 (sec). Leaf size: 75
DSolve[(x*y'''[x]-y''[x])^2==(y'''[x])^2+1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 x^3}{6}-\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2 \\ y(x)\to \frac {c_1 x^3}{6}+\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2 \\ \end{align*}