29.23.4 problem 634

Internal problem ID [5226]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 634
Date solved : Monday, January 27, 2025 at 10:23:52 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.166 (sec). Leaf size: 105

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)
 

\[ y \left (x \right ) = x^{\frac {-2 a +2}{2+a}} {\mathrm e}^{\frac {\left (-a +1\right ) \operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}}-x^{-\frac {6 a}{2+a}} {\mathrm e}^{-\frac {2 \left (\textit {\_Z} a +3 c_{1} a -\textit {\_Z} \right )}{2+a}}-3 x^{-\frac {3 a}{2+a}} {\mathrm e}^{-\frac {\textit {\_Z} a +3 c_{1} a -\textit {\_Z}}{2+a}}-3\right )-3 c_{1} a}{2+a}}+x \]

✓ Solution by Mathematica

Time used: 0.217 (sec). Leaf size: 54

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

\[ \text {Solve}\left [\frac {1}{3} (a-1) \log \left (\frac {y(x)^2}{x^2}+\frac {y(x)}{x}+1\right )+\frac {1}{3} (a+2) \log \left (1-\frac {y(x)}{x}\right )=-a \log (x)+c_1,y(x)\right ] \]