60.1.141 problem 142

Internal problem ID [10155]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 142
Date solved : Monday, January 27, 2025 at 06:30:18 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.852 (sec). Leaf size: 76

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

\begin{align*} y(x)\to \frac {\int _1^xe^{a K[1]+2} K[1]^2dK[1]+x^3 \left (-e^{a x+2}\right )+c_1}{x \left (\int _1^xe^{a K[1]+2} K[1]^2dK[1]+c_1\right )} \\ y(x)\to \frac {1}{x} \\ \end{align*}