Internal problem ID [10302]
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 5.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
Solve \begin {gather*} \boxed {\left (y^{\prime }-x y^{\prime \prime }\right )^{2}-1-\left (y^{\prime \prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 63
dsolve((diff(y(x),x)-x*diff(y(x),x$2))^2=1+diff(y(x),x$2)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x \sqrt {-x^{2}+1}}{2}+\frac {\arcsin \relax (x )}{2}+c_{1} \\ y \relax (x ) = -\frac {x \sqrt {-x^{2}+1}}{2}-\frac {\arcsin \relax (x )}{2}+c_{1} \\ y \relax (x ) = \frac {x^{2} \sqrt {c_{1}^{2}-1}}{2}+x c_{1}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.124 (sec). Leaf size: 58
DSolve[(y'[x]-x*y''[x])^2==1+(y''[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 x^2}{2}-\sqrt {1+c_1{}^2} x+c_2 \\ y(x)\to \frac {c_1 x^2}{2}+\sqrt {1+c_1{}^2} x+c_2 \\ \end{align*}