31.6.18 problem 18

Internal problem ID [5767]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 18
Date solved : Monday, January 27, 2025 at 01:16:01 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.131 (sec). Leaf size: 77

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

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-4 x -1}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-4 x -1}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {4 x^{2}+\left (-8 c_{1} -4\right ) x +4 c_{1}^{2}-1}}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 x^{2}+\left (-8 c_{1} -4\right ) x +4 c_{1}^{2}-1}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.242 (sec). Leaf size: 69

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

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {4 x^2-4 (1+4 c_1) x-1+16 c_1{}^2} \\ y(x)\to \frac {1}{2} \sqrt {4 x^2-4 (1+4 c_1) x-1+16 c_1{}^2} \\ \end{align*}