69.1.70 problem 98

Internal problem ID [14223]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 98
Date solved : Tuesday, January 28, 2025 at 06:22:32 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y&=x y^{\prime }-\frac {1}{{y^{\prime }}^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 72

dsolve(y(x)=x*diff(y(x),x)-1/diff(y(x),x)^2,y(x), singsol=all)
 

\begin{align*} y &= \frac {3 \,2^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}}}{2} \\ y &= -\frac {3 \,2^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {3 \,2^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4} \\ y &= c_{1} x -\frac {1}{c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 71

DSolve[y[x]==x*D[y[x],x]-1/(D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 x-\frac {1}{c_1{}^2} \\ y(x)\to -3 \left (-\frac {1}{2}\right )^{2/3} x^{2/3} \\ y(x)\to -\frac {3 x^{2/3}}{2^{2/3}} \\ y(x)\to \frac {3 \sqrt [3]{-1} x^{2/3}}{2^{2/3}} \\ \end{align*}