58.5.31 problem 31

Internal problem ID [14625]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.3 (Linear equations). Exercises page 56
Problem number : 31
Date solved : Thursday, October 02, 2025 at 09:44:39 AM
CAS classification : [[_linear, `class A`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to \frac {e^{-\frac {b x}{a}} \left (k e^{x \left (\frac {b}{a}+\lambda \right )}+c_1 (a \lambda +b)\right )}{a \lambda +b} \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{1} e^{- \frac {b x}{a}} - \frac {k e^{- \lambda _{} x}}{a \lambda _{} - b} \]