90.25.20 problem 21

Internal problem ID [25401]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 379
Problem number : 21
Date solved : Friday, October 03, 2025 at 12:01:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-\left (a +b \right ) y^{\prime }+a b y&=f \left (t \right ) \end{align*}

Using Laplace method

✓ Maple. Time used: 0.093 (sec). Leaf size: 76

ode:=diff(diff(y(t),t),t)-(a+b)*diff(y(t),t)+a*b*y(t) = f(t); 
dsolve(ode,y(t),method='laplace');
 

\[ y = \frac {\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{a \left (t -\textit {\_U1} \right )}d \textit {\_U1} -\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{b \left (t -\textit {\_U1} \right )}d \textit {\_U1} +{\mathrm e}^{a t} \left (-y \left (0\right ) b +y^{\prime }\left (0\right )\right )+{\mathrm e}^{b t} \left (y \left (0\right ) a -y^{\prime }\left (0\right )\right )}{-b +a} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 85

ode=D[y[t],{t,2}]-(a+b)*D[y[t],t]+a*b*y[t]==f[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{b t} \int _1^t-\frac {e^{-b K[1]} f(K[1])}{a-b}dK[1]+e^{a t} \int _1^t\frac {e^{-a K[2]} f(K[2])}{a-b}dK[2]+c_2 e^{a t}+c_1 e^{b t} \end{align*}

✓ Sympy. Time used: 0.587 (sec). Leaf size: 42

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

\[ y{\left (t \right )} = \left (C_{1} + \frac {\int f{\left (t \right )} e^{- a t}\, dt}{a - b}\right ) e^{a t} + \left (C_{2} - \frac {\int f{\left (t \right )} e^{- b t}\, dt}{a - b}\right ) e^{b t} \]