62.33.5 problem Ex 5

Internal problem ID [12992]
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
Date solved : Tuesday, January 28, 2025 at 04:46:32 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.198 (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 &= \frac {\sqrt {-x^{2}+1}\, x}{2}+\frac {\arcsin \left (x \right )}{2}+c_{1} \\ y &= -\frac {\sqrt {-x^{2}+1}\, x}{2}-\frac {\arcsin \left (x \right )}{2}+c_{1} \\ y &= \frac {\sqrt {c_{1}^{2}-1}\, x^{2}}{2}+c_{1} x +c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.148 (sec). Leaf size: 58

DSolve[(D[y[x],x]-x*D[y[x],{x,2}])^2==1+(D[y[x],{x,2}])^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*}