61.3.14 problem 14

Internal problem ID [12019]
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 : 14
Date solved : Sunday, March 30, 2025 at 10:16:58 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 96

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

\[ y = \frac {a c_1 \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {2 \mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+\lambda -\mu }{\left (-\mu +\lambda \right ) \left (c_1 \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+{\mathrm e}^{\lambda x}\right )} \]

✓ Mathematica. Time used: 2.271 (sec). Leaf size: 165

ode=D[y[x],x]==-\[Lambda]*Exp[\[Lambda]*x]*y[x]^2+a*Exp[\[Mu]*x]*y[x]-a*Exp[(\[Mu]-\[Lambda])*x]; 
ic={}; 
DSolve[{ode,ic},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 )}{\lambda \left (-\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 }\right )} \\ y(x)\to e^{\lambda (-x)} \\ \end{align*}

✗ Sympy

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

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