54.2.19 problem 22

Internal problem ID [8551]
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
Date solved : Monday, January 27, 2025 at 04:14:03 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 91

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

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

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 102

DSolve[(D[y[x],x]+1)^2*(y[x]-D[y[x],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 -x+\frac {3 i \left (\sqrt {3}+i\right ) x^{2/3}}{2\ 2^{2/3}} \\ y(x)\to -x-\frac {3 \left (1+i \sqrt {3}\right ) x^{2/3}}{2\ 2^{2/3}} \\ \end{align*}