46.7.4 problem 12

Internal problem ID [9645]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.4.1 DERIVATIVES OF A TRANSFORM. Page 309
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 06:21:48 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.113 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)+y(t) = sin(t); 
ic:=[y(0) = 1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right ) \left (-2+t \right )}{2} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 57

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

\begin{align*} y(t)&\to \cos (t) \left (-\int _1^0-\sin ^2(K[1])dK[1]\right )+\cos (t) \int _1^t-\sin ^2(K[1])dK[1]-\frac {5 \sin (t)}{8}-\frac {1}{8} \sin (3 t)+\cos (t) \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 15

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

\[ y{\left (t \right )} = \left (1 - \frac {t}{2}\right ) \cos {\left (t \right )} - \frac {\sin {\left (t \right )}}{2} \]