61.20.8 problem 41

Internal problem ID [12310]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-2. Equations containing arbitrary functions and their derivatives.
Problem number : 41
Date solved : Tuesday, January 28, 2025 at 02:34:48 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 28

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

\[ y = -\tan \left (-\sqrt {a}\, \left (\int f \,{\mathrm e}^{g \left (x \right )}d x \right )+c_{1} \right ) \sqrt {a}\, {\mathrm e}^{g \left (x \right )} \]

✓ Solution by Mathematica

Time used: 0.998 (sec). Leaf size: 35

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

\begin{align*} y(x)\to -i \sqrt {a} e^{g(x)} \\ y(x)\to -i \sqrt {a} e^{g(x)} \\ \end{align*}