Internal problem ID [9876]
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]
Solve \begin {gather*} \boxed {x y^{\prime }-f \relax (x ) \left (y+a \ln \relax (x )\right )^{2}+a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 24
dsolve(x*diff(y(x),x)=f(x)*(y(x)+a*ln(x))^2-a,y(x), singsol=all)
\[ y \relax (x ) = -a \ln \relax (x )+\frac {1}{c_{1}-\left (\int \frac {f \relax (x )}{x}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.31 (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*}