28.1.77 problem 80

Internal problem ID [4383]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 80
Date solved : Monday, January 27, 2025 at 09:09:45 AM
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: 0.056 (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 \left (x \right ) &= \frac {3 \,2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}}}{2} \\ y \left (x \right ) &= -\frac {3 \,2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 \,2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4} \\ y \left (x \right ) &= c_{1} x +\frac {1}{c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.012 (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*}