90.25.18 problem 19

Internal problem ID [25399]
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 : 19
Date solved : Friday, October 03, 2025 at 12:01:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method

✓ Maple. Time used: 0.091 (sec). Leaf size: 63

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

\[ y = y \left (0\right ) \cosh \left (a t \right )+\frac {2 y^{\prime }\left (0\right ) \sinh \left (a t \right )+\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{a \left (t -\textit {\_U1} \right )}d \textit {\_U1} -\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{-a \left (t -\textit {\_U1} \right )}d \textit {\_U1}}{2 a} \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 78

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

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

✓ Sympy. Time used: 0.427 (sec). Leaf size: 42

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

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