Internal problem ID [3314]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 2
Problem number: 50.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-x f \left (x \right ) y-y^{2}=f \left (x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
dsolve(diff(y(x),x) = f(x)+x*f(x)*y(x)+y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}}{c_{1} -\left (\int {\mathrm e}^{\int \frac {f \left (x \right ) x^{2}-2}{x}d x}d x \right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 0.756 (sec). Leaf size: 111
DSolve[y'[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*}