62.33.2 problem Ex 2

Internal problem ID [12989]
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
Date solved : Tuesday, January 28, 2025 at 04:46:27 AM
CAS classification : [[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.214 (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 &= \frac {\left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{6}+c_{1} x +\frac {x \arcsin \left (x \right )}{2}+c_{2} \\ y &= -\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 &= \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: 0.164 (sec). Leaf size: 75

DSolve[(x*D[y[x],{x,3}]-D[y[x],{x,2}])^2==(D[y[x],{x,3}])^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*}