46.7.9 problem 36

Internal problem ID [9650]
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 : 36
Date solved : Tuesday, September 30, 2025 at 06:21:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.110 (sec). Leaf size: 18

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

\[ y = -\frac {\left (\cos \left (t \right ) t -\sin \left (t \right )\right ) \left (t +2\right )}{4} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 91

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

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

✓ Sympy. Time used: 0.087 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = \left (\frac {t}{4} + \frac {1}{2}\right ) \sin {\left (t \right )} + \left (- \frac {t^{2}}{4} - \frac {t}{2}\right ) \cos {\left (t \right )} \]