61.3.13 problem 13

Internal problem ID [12097]
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 : 13
Date solved : Tuesday, January 28, 2025 at 12:22:11 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 97

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

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

✓ Solution by Mathematica

Time used: 2.311 (sec). Leaf size: 166

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

\begin{align*} y(x)\to -\frac {e^{\lambda (-x)} \left (-\lambda \left (-\frac {a e^{\mu x}}{\mu }\right )^{\lambda /\mu } \Gamma \left (-\frac {\lambda }{\mu },-\frac {a e^{x \mu }}{\mu }\right )+\mu e^{\frac {a e^{\mu x}}{\mu }}+c_1 \lambda e^{\frac {\lambda }{\mu }+1} \left (e^{\mu x}\right )^{\lambda /\mu }\right )}{-\left (-\frac {a e^{\mu x}}{\mu }\right )^{\lambda /\mu } \Gamma \left (-\frac {\lambda }{\mu },-\frac {a e^{x \mu }}{\mu }\right )+c_1 e^{\frac {\lambda }{\mu }+1} \left (e^{\mu x}\right )^{\lambda /\mu }} \\ y(x)\to \lambda \left (-e^{\lambda (-x)}\right ) \\ \end{align*}