29.23.5 problem 635

Internal problem ID [5227]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 635
Date solved : Monday, January 27, 2025 at 10:24:16 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 59

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

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

✓ Solution by Mathematica

Time used: 4.739 (sec). Leaf size: 87

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

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