90.25.19 problem 20

Internal problem ID [25400]
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 : 20
Date solved : Friday, October 03, 2025 at 12:01:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 a y^{\prime }+a^{2} y&=f \left (t \right ) \end{align*}

Using Laplace method

✓ Maple. Time used: 0.070 (sec). Leaf size: 45

ode:=diff(diff(y(t),t),t)-2*a*diff(y(t),t)+a^2*y(t) = f(t); 
dsolve(ode,y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 59

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

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

✓ Sympy. Time used: 0.934 (sec). Leaf size: 34

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

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