Internal problem ID [3309]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 2
Problem number: 45.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (x^{2}+1\right ) y-y^{2}=2 x} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 67
dsolve(diff(y(x),x) = 2*x-(x^2+1)*y(x)+y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-x^{2} \left (\int {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}d x \right )+c_{1} x^{2}+{\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}-\left (\int {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}d x \right )+c_{1}}{c_{1} -\left (\int {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.337 (sec). Leaf size: 58
DSolve[y'[x]==2 x-(1+x^2)y[x]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\frac {x^3}{3}+x}}{-\int _1^xe^{\frac {K[1]^3}{3}+K[1]}dK[1]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}