55.19.12 problem 12

Internal problem ID [13497]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 12
Date solved : Wednesday, October 01, 2025 at 03:32:59 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 86

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

\[ y = -\frac {c_1 \,{\mathrm e}^{-2 \lambda x +a \int {\mathrm e}^{\lambda x} f \left (x \right )d x}+{\mathrm e}^{-\lambda x} \left (\lambda \int {\mathrm e}^{-\lambda x +a \int {\mathrm e}^{\lambda x} f \left (x \right )d x}d x c_1 +\lambda ^{2}\right )}{a \left (\int {\mathrm e}^{-\lambda x +a \int {\mathrm e}^{\lambda x} f \left (x \right )d x}d x c_1 +\lambda \right )} \]

✓ Mathematica. Time used: 1.496 (sec). Leaf size: 166

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

\begin{align*} y(x)&\to -\frac {\lambda e^{-2 \lambda x} \left (\exp \left (-\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )+e^{\lambda x} \int _1^{e^{x \lambda }}\frac {\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )}{K[2]^2}dK[2]+c_1 e^{\lambda x}\right )}{a \left (\int _1^{e^{x \lambda }}\frac {\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )}{K[2]^2}dK[2]+c_1\right )}\\ y(x)&\to -\frac {\lambda e^{\lambda (-x)}}{a} \end{align*}

✗ Sympy

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

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