53.3.15 problem 17

Internal problem ID [8477]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 99. Clairaut equation. EXERCISES Page 320
Problem number : 17
Date solved : Monday, January 27, 2025 at 04:07:04 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 64

dsolve(x*diff(y(x),x)^3-y(x)*diff(y(x),x)^2+1=0,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: 0.013 (sec). Leaf size: 69

DSolve[x*(D[y[x],x])^3-y[x]*(D[y[x],x])^2+1==0,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*}