Internal problem ID [10299]
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: 0.047 (sec). Leaf size: 97
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 \relax (x ) = -\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{6}+\frac {x \arcsin \relax (x )}{2}+\frac {\sqrt {-x^{2}+1}}{2}+x c_{1}+c_{2} \\ y \relax (x ) = \frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{6}-\frac {x \arcsin \relax (x )}{2}-\frac {\sqrt {-x^{2}+1}}{2}+x c_{1}+c_{2} \\ y \relax (x ) = \frac {x^{3} \sqrt {c_{1}^{2}-1}}{6}+\frac {x^{2} c_{1}}{2}+c_{2} x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.13 (sec). Leaf size: 73
DSolve[(x*y'''[x]-y''[x])^2==(y'''[x])^2+1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} x \left (c_1 x^2-3 \sqrt {1+c_1{}^2} x+6 c_3\right )+c_2 \\ y(x)\to \frac {1}{6} x \left (c_1 x^2+3 \sqrt {1+c_1{}^2} x+6 c_3\right )+c_2 \\ \end{align*}