Internal problem ID [10619]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing
arbitrary functions (but not containing their derivatives).
Problem number: 26.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {y^{\prime } x -f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}=-a} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
dsolve(x*diff(y(x),x)=f(x)*(y(x)+a*ln(x))^2-a,y(x), singsol=all)
\[ y \left (x \right ) = -a \ln \left (x \right )+\frac {1}{c_{1} -\left (\int \frac {f \left (x \right )}{x}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.48 (sec). Leaf size: 42
DSolve[x*y'[x]==f[x]*(y[x]+a*Log[x])^2-a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -a \log (x)+\frac {1}{-\int _1^x\frac {f(K[2])}{K[2]}dK[2]+c_1} \\ y(x)\to -a \log (x) \\ \end{align*}