2.3.6 Problem 6

Internal problem ID [13286]
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 : 6
Date solved : Wednesday, December 31, 2025 at 12:56:33 PM
CAS classification : [_Riccati]

2.3.6.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

0.059 (sec)

Entering first order ode riccati guess solver

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

This is a Riccati ODE. Comparing the above ODE to solve with the Riccati standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that

\begin{align*} f_0(x) & =-a \,{\mathrm e}^{\lambda x} b -b^{2}\\ f_1(x) & ={\mathrm e}^{\lambda x} a\\ f_2(x) &=1 \end{align*}

Using trial and error, the following particular solution was found

\[ y_p = b \]

Since a particular solution is known, then the general solution is given by

\begin{align*} y &= y_p + \frac { \phi (x) }{ c_1 - \int { \phi (x) f_2 \,dx} } \end{align*}

Where

\begin{align*} \phi (x) &= e^{ \int 2 f_2 y_p + f_1 \,dx } \end{align*}

Evaluating the above gives the general solution as

\[ y = b +\frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }+2 b x}}{c_1 -\int {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }+2 b x}d x} \]

Summary of solutions found

\begin{align*} y &= b +\frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }+2 b x}}{c_1 -\int {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }+2 b x}d x} \\ \end{align*}

2.3.6.2 ✓ Maple. Time used: 0.001 (sec). Leaf size: 73

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

\[ y = \frac {b c_1 -b \int {\mathrm e}^{\frac {2 \lambda x b +{\mathrm e}^{\lambda x} a}{\lambda }}d x +{\mathrm e}^{\frac {2 \lambda x b +{\mathrm e}^{\lambda x} a}{\lambda }}}{-\int {\mathrm e}^{\frac {2 \lambda x b +{\mathrm e}^{\lambda x} a}{\lambda }}d x +c_1} \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying Riccati 
trying Riccati sub-methods: 
   <- Riccati particular case Kamke (b) successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \,{\mathrm e}^{\lambda x} y \left (x \right )-a b \,{\mathrm e}^{\lambda x}-b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \,{\mathrm e}^{\lambda x} y \left (x \right )-a b \,{\mathrm e}^{\lambda x}-b^{2} \end {array} \]

2.3.6.3 ✓ Mathematica. Time used: 0.322 (sec). Leaf size: 115

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

\begin{align*} y(x)&\to \frac {b \left (-2 \lambda e^{\frac {a e^{\lambda x}}{\lambda }} \left (-\frac {a e^{\lambda x}}{\lambda }\right )^{\frac {2 b}{\lambda }}+2 b \Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda (-1)^{b/\lambda }\right )}{2 b \Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda (-1)^{b/\lambda }}\\ y(x)&\to b \end{align*}

2.3.6.4 ✗ Sympy

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

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