90.25.17 problem 18

Internal problem ID [25398]
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 : 18
Date solved : Friday, October 03, 2025 at 12:01:07 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: 0.175 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 69

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 \cos (a t) \int _1^t-\frac {f(K[1]) \sin (a K[1])}{a}dK[1]+\sin (a t) \int _1^t\frac {\cos (a K[2]) f(K[2])}{a}dK[2]+c_1 \cos (a t)+c_2 \sin (a t) \end{align*}

✓ Sympy. Time used: 0.495 (sec). Leaf size: 53

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)
 

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