14.17.2 problem 20

Internal problem ID [2680]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 20
Date solved : Sunday, March 30, 2025 at 12:13:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 25

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

\[ y(t)\to \frac {1}{4} \left ((t+8) \sin (t)-\left (t^2-4\right ) \cos (t)\right ) \]

✓ Sympy. Time used: 0.138 (sec). Leaf size: 20

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

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