60.1.20 problem 20

Internal problem ID [10034]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 20
Date solved : Monday, January 27, 2025 at 06:18:56 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.299 (sec). Leaf size: 58

DSolve[D[y[x],x] - y[x]^2 +(x^2 + 1)*y[x] - 2*x==0,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*}