29.2.25 problem 50

Internal problem ID [4658]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 2
Problem number : 50
Date solved : Monday, January 27, 2025 at 09:29:48 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 69

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

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

✓ Solution by Mathematica

Time used: 0.742 (sec). Leaf size: 111

DSolve[D[y[x],x]==f[x]+x f[x] y[x]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

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