29.9.25 problem 265

Internal problem ID [4865]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 9
Problem number : 265
Date solved : Monday, January 27, 2025 at 09:45:33 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x^{2} y^{\prime }+2+a x \left (1-y x \right )-x^{2} y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 52

dsolve(x^2*diff(y(x),x)+2+a*x*(1-x*y(x))-x^2*y(x)^2 = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-\left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+c_{1}}{x \left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x}+c_{1} \right )} \]

✓ Solution by Mathematica

Time used: 0.342 (sec). Leaf size: 78

DSolve[x^2 D[y[x],x]+2+a x(1-x y[x])-x^2 y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

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