60.2.413 problem 990

Internal problem ID [11000]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 990
Date solved : Monday, January 27, 2025 at 10:39:58 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=-F \left (x \right ) \left (-y^{2}+2 x^{2} y+1-x^{4}\right )+2 x \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 44

dsolve(diff(y(x),x) = -F(x)*(-y(x)^2+2*x^2*y(x)+1-x^4)+2*x,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.261 (sec). Leaf size: 67

DSolve[D[y[x],x] == 2*x - F[x]*(1 - x^4 + 2*x^2*y[x] - y[x]^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\exp \left (\int _1^x2 F(K[5])dK[5]\right )}{-\int _1^x\exp \left (\int _1^{K[6]}2 F(K[5])dK[5]\right ) F(K[6])dK[6]+c_1}+x^2+1 \\ y(x)\to x^2+1 \\ \end{align*}