61.19.17 problem 17

Internal problem ID [12286]
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 : 17
Date solved : Tuesday, January 28, 2025 at 02:09:13 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.094 (sec). Leaf size: 59

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

\[ y = -\frac {\left (a^{2}+\tanh \left (\frac {\sqrt {a^{2} \left (a^{2}-4 b \right )}\, \left (a \left (\int fd x \right )+c_{1} \right )}{2 a^{2}}\right ) \sqrt {a^{2} \left (a^{2}-4 b \right )}\right ) {\mathrm e}^{-\lambda x}}{2 a} \]

✓ Solution by Mathematica

Time used: 0.669 (sec). Leaf size: 87

DSolve[D[y[x],x]==Exp[\[Lambda]*x]*f[x]*y[x]^2+(a*f[x]-\[Lambda])*y[x]+b*Exp[-\[Lambda]*x]*f[x],y[x],x,IncludeSingularSolutions -> True]
 

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