82.5.3 problem Ex. 3

Internal problem ID [18739]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 20
Problem number : Ex. 3
Date solved : Tuesday, January 28, 2025 at 12:13:57 PM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} a^{2}-2 y x -y^{2}-\left (x +y\right )^{2} y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 111

dsolve((a^2-2*x*y(x)-y(x)^2)-(x+y(x))^2*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.501 (sec). Leaf size: 109

DSolve[(a^2-2*x*y[x]-y[x]^2)-(x+y[x])^2*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x+\sqrt [3]{3 a^2 x+x^3+3 c_1} \\ y(x)\to -x+\frac {1}{2} i \left (\sqrt {3}+i\right ) \sqrt [3]{3 a^2 x+x^3+3 c_1} \\ y(x)\to -x-\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{3 a^2 x+x^3+3 c_1} \\ \end{align*}