61.3.5 problem 5

Internal problem ID [12010]
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 : Wednesday, March 05, 2025 at 03:50:26 PM
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*}

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

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

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

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

ode=D[y[x],x]==y[x]^2+b*y[x]+a*(\[Lambda]-b)*Exp[\[Lambda]*x]-a^2*Exp[2*\[Lambda]*x]; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
cg = symbols("cg") 
y = Function("y") 
ode = Eq(a**2*exp(2*cg*x) - a*(-b + cg)*exp(cg*x) - b*y(x) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a**2*exp(2*cg*x) + a*b*exp(cg*x) - a*cg*exp(cg*x) - b*y(x) - y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method