Internal problem ID [6130]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 22.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }+1\right )^{2} \left (-x y^{\prime }+y\right )-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.254 (sec). Leaf size: 106
dsolve((diff(y(x),x)+1)^2*(y(x)-diff(y(x),x)*x)=1,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{2}-x \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}-\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}-x \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}+\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}-x \\ y \relax (x ) = \frac {\left (c_{1}^{3}+2 c_{1}^{2}+c_{1}\right ) x}{\left (c_{1}+1\right )^{2}}+\frac {1}{\left (c_{1}+1\right )^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 82
DSolve[(y'[x]+1)^2*(y[x]-y'[x]*x)==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x+\frac {1}{(1+c_1){}^2} \\ y(x)\to \frac {3 x^{2/3}}{2^{2/3}}-x \\ y(x)\to 3 \left (-\frac {1}{2}\right )^{2/3} x^{2/3}-x \\ y(x)\to \frac {1}{2} \left (-3 \sqrt [3]{-2} x^{2/3}-2 x\right ) \\ \end{align*}