61.3.17 problem 17

Internal problem ID [12022]
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 : 17
Date solved : Wednesday, March 05, 2025 at 03:52:01 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 79

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

\[ y = \frac {{\mathrm e}^{-\lambda x} \left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\left (b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+\left (\lambda +\mu \right ) c_{1} \right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2} \left (\lambda +\mu \right )}\right )-b^{2}\right )}{2 a b} \]

✓ Mathematica. Time used: 3.004 (sec). Leaf size: 349

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

\begin{align*} y(x)\to \frac {e^{\lambda (-x)} \left (b^2 e^{x (\lambda +\mu )} \left (\pi +i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}-1\right )\right )-b (\lambda +\mu ) \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \left (\pi -i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}+1\right )\right )-4 a c e^{x (\lambda +\mu )} \left (\pi +i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}-1\right )\right )\right )}{2 a (\lambda +\mu ) \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \left (\pi -i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}+1\right )\right )} \\ y(x)\to \frac {e^{\lambda (-x)} \left (-(\lambda +\mu ) e^{-x (\lambda +\mu )} \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \tanh \left (\frac {1}{2} \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}\right )-b\right )}{2 a} \\ \end{align*}

✗ Sympy

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

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