61.3.19 problem 19

Internal problem ID [12103]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3. Equations Containing Exponential Functions
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 12:23:12 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 44

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

\[ y = \frac {\left ({\mathrm e}^{x \left (\lambda +\mu \right )} c_{1} b \mu +b \,{\mathrm e}^{\lambda x}-c_{1} \mu ^{2}\right ) {\mathrm e}^{-\mu x}}{c_{1} \mu +{\mathrm e}^{-\mu x}} \]

✓ Solution by Mathematica

Time used: 0.926 (sec). Leaf size: 40

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

\begin{align*} y(x)\to b e^{\lambda x}+\frac {\mu }{-e^{\mu x}+c_1 \mu } \\ y(x)\to b e^{\lambda x} \\ \end{align*}