52.7.1 problem 9

Internal problem ID [8356]
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 : 9
Date solved : Wednesday, March 05, 2025 at 05:35:47 AM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.585 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 28

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

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

✓ Sympy. Time used: 0.170 (sec). Leaf size: 27

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

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