61.19.1 problem 1

Internal problem ID [12270]
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 : 1
Date solved : Tuesday, January 28, 2025 at 02:02:39 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.345 (sec). Leaf size: 166

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

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}(-2 a-f(K[1]))dK[1]\right ) (a+f(K[2])+y(x))}{a-y(x)}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x(-2 a-f(K[1]))dK[1]\right )}{(K[3]-a)^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}(-2 a-f(K[1]))dK[1]\right ) (a+f(K[2])+K[3])}{(a-K[3])^2}+\frac {\exp \left (-\int _1^{K[2]}(-2 a-f(K[1]))dK[1]\right )}{a-K[3]}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]