61.19.26 problem 26

Internal problem ID [12295]
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
Date solved : Tuesday, January 28, 2025 at 07:54:15 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 24

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

\[ y = -a \ln \left (x \right )+\frac {1}{c_{1} -\int \frac {f}{x}d x} \]

✓ Solution by Mathematica

Time used: 0.294 (sec). Leaf size: 42

DSolve[x*D[y[x],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*}