29.20.16 problem 561

Internal problem ID [5157]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 20
Problem number : 561
Date solved : Monday, January 27, 2025 at 10:16:23 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.181 (sec). Leaf size: 60

dsolve(x*(x-a*y(x))*diff(y(x),x) = y(x)*(y(x)-a*x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.171 (sec). Leaf size: 36

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

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