8.16.4 problem 18

Internal problem ID [2674]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 05:49:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=t^{2} \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: 0.117 (sec). Leaf size: 27

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

\[ y = \frac {\left (-2 t^{3}+3 t \right ) \cos \left (t \right )}{12}+\frac {\sin \left (t \right ) \left (t^{2}-1\right )}{4} \]

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 31

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

\begin{align*} y(t)&\to \frac {1}{12} \left (\left (3 t-2 t^3\right ) \cos (t)+3 \left (t^2-1\right ) \sin (t)\right ) \end{align*}

✓ Sympy. Time used: 0.143 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*sin(t) + y(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)
 

\[ y{\left (t \right )} = \left (\frac {t^{2}}{4} - \frac {1}{4}\right ) \sin {\left (t \right )} + \left (- \frac {t^{3}}{6} + \frac {t}{4}\right ) \cos {\left (t \right )} \]