29.2.20 problem 45

Internal problem ID [4653]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 2
Problem number : 45
Date solved : Monday, January 27, 2025 at 09:29:33 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 67

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

\[ y \left (x \right ) = \frac {-x^{2} \left (\int {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}d x \right )+c_{1} x^{2}-\int {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}d x +{\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}+c_{1}}{c_{1} -\int {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}}d x} \]

✓ Solution by Mathematica

Time used: 0.483 (sec). Leaf size: 58

DSolve[D[y[x],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*}