61.3.5 problem 5

Internal problem ID [12089]
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 : 5
Date solved : Tuesday, January 28, 2025 at 12:21:40 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 80

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

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

✓ Solution by Mathematica

Time used: 1.612 (sec). Leaf size: 191

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

\begin{align*} y(x)\to \frac {-2^{b/\lambda } \left (b-a e^{\lambda x}\right ) \left (\frac {a e^{\lambda x}}{\lambda }\right )^{b/\lambda } L_{-\frac {b}{\lambda }}^{\frac {b}{\lambda }}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+a e^{\lambda x} \left (2^{\frac {b+\lambda }{\lambda }} \left (\frac {a e^{\lambda x}}{\lambda }\right )^{b/\lambda } L_{-\frac {b+\lambda }{\lambda }}^{\frac {b+\lambda }{\lambda }}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+c_1\right )}{2^{b/\lambda } \left (\frac {a e^{\lambda x}}{\lambda }\right )^{b/\lambda } L_{-\frac {b}{\lambda }}^{\frac {b}{\lambda }}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+c_1} \\ y(x)\to a e^{\lambda x} \\ \end{align*}