Internal problem ID [3374]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 634.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (x^{2}+y x +a y^{2}\right ) y^{\prime }-a \,x^{2}-y x -y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 216
dsolve((x^2+x*y(x)+a*y(x)^2)*diff(y(x),x) = a*x^2+x*y(x)+y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-\frac {3 c_{1} a +\RootOf \left ({\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{-\frac {6 c_{1} a}{a +2}} {\mathrm e}^{-\frac {2 a \textit {\_Z}}{a +2}} x^{-\frac {6 a}{a +2}} {\mathrm e}^{\frac {2 \textit {\_Z}}{a +2}}-3 \,{\mathrm e}^{-\frac {3 c_{1} a}{a +2}} {\mathrm e}^{-\frac {a \textit {\_Z}}{a +2}} x^{-\frac {3 a}{a +2}} {\mathrm e}^{\frac {\textit {\_Z}}{a +2}}-3\right ) a +3 a \ln \relax (x )-\RootOf \left ({\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{-\frac {6 c_{1} a}{a +2}} {\mathrm e}^{-\frac {2 a \textit {\_Z}}{a +2}} x^{-\frac {6 a}{a +2}} {\mathrm e}^{\frac {2 \textit {\_Z}}{a +2}}-3 \,{\mathrm e}^{-\frac {3 c_{1} a}{a +2}} {\mathrm e}^{-\frac {a \textit {\_Z}}{a +2}} x^{-\frac {3 a}{a +2}} {\mathrm e}^{\frac {\textit {\_Z}}{a +2}}-3\right )}{a +2}} x +x \]
✓ Solution by Mathematica
Time used: 0.21 (sec). Leaf size: 54
DSolve[(x^2+x y[x]+a y[x]^2)y'[x]==a x^2+x y[x]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{3} (a-1) \log \left (\frac {y(x)^2}{x^2}+\frac {y(x)}{x}+1\right )+\frac {1}{3} (a+2) \log \left (1-\frac {y(x)}{x}\right )=-a \log (x)+c_1,y(x)\right ] \]