61.3.12 problem 12

Internal problem ID [12096]
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 : 12
Date solved : Tuesday, January 28, 2025 at 12:22:09 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 79

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

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

✓ Solution by Mathematica

Time used: 2.538 (sec). Leaf size: 25

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

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