46.7.2 problem 10

Internal problem ID [9643]
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 : 10
Date solved : Tuesday, September 30, 2025 at 06:21:47 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-y&=t \,{\mathrm e}^{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: 0.129 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 22

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

\begin{align*} y(t)&\to e^t \int _0^tK[1] \sin (K[1])dK[1] \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 14

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

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