76.22.1 problem 14

Internal problem ID [17711]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 14
Date solved : Monday, March 31, 2025 at 04:25:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+w^{2} y&=g \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.269 (sec). Leaf size: 32

ode:=diff(diff(y(t),t),t)+w^2*y(t) = g(t); 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {-\int _{0}^{t}g \left (\textit {\_U1} \right ) \sin \left (w \left (-t +\textit {\_U1} \right )\right )d \textit {\_U1} +\sin \left (w t \right )}{w} \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 114

ode=D[y[t],{t,2}]+w^2*y[t]==g[t]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \sin (t w) \left (-\int _1^0\frac {\cos (w K[2]) g(K[2])}{w}dK[2]\right )+\sin (t w) \int _1^t\frac {\cos (w K[2]) g(K[2])}{w}dK[2]-\cos (t w) \int _1^0-\frac {g(K[1]) \sin (w K[1])}{w}dK[1]+\cos (t w) \int _1^t-\frac {g(K[1]) \sin (w K[1])}{w}dK[1]+\frac {\sin (t w)}{w} \]

✓ Sympy. Time used: 0.813 (sec). Leaf size: 94

from sympy import * 
t = symbols("t") 
w = symbols("w") 
y = Function("y") 
ode = Eq(w**2*y(t) - g(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {i \int g{\left (t \right )} e^{- i t w}\, dt}{2 w} + \frac {i \int \limits ^{0} g{\left (t \right )} e^{- i t w}\, dt}{2 w} - \frac {i}{2 w}\right ) e^{i t w} + \left (\frac {i \int g{\left (t \right )} e^{i t w}\, dt}{2 w} - \frac {i \int \limits ^{0} g{\left (t \right )} e^{i t w}\, dt}{2 w} + \frac {i}{2 w}\right ) e^{- i t w} \]