66.21.4 problem 4

Internal problem ID [16261]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.6. page 624
Problem number : 4
Date solved : Thursday, October 02, 2025 at 10:44:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+3 y&=\left (1-\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\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: 0.623 (sec). Leaf size: 178

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+3*y(t) = (1-Heaviside(t-2))*exp(-1/10*t+1/5)*sin(t-2); 
ic:=[y(0) = 1, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {10 \left (\left (4400+10505 i\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-1590+1564 i\right ) \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{\frac {1}{5}-2 i-\frac {t}{2}}}{471691}+\frac {10 \left (\left (4400-10505 i\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-1590-1564 i\right ) \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{\frac {1}{5}+2 i-\frac {t}{2}}}{471691}+\left (\frac {4000}{42881}-\frac {9550 i}{42881}\right ) \left (-1+\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (\frac {4000}{42881}+\frac {9550 i}{42881}\right ) \left (-1+\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}-\frac {31800 \left (\left (\sin \left (\sqrt {11}\right ) \sqrt {11}+\frac {440 \cos \left (\sqrt {11}\right )}{159}\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-\sin \left (\frac {\sqrt {11}\, t}{2}\right ) \left (\sqrt {11}\, \cos \left (\sqrt {11}\right )-\frac {440 \sin \left (\sqrt {11}\right )}{159}\right )\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{1-\frac {t}{2}}}{471691}+\left (\frac {5 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{11}+\cos \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}} \]

✓ Mathematica. Time used: 8.294 (sec). Leaf size: 1335

ode=D[y[t],{t,2}]+D[y[t],t]+8*y[t]==(1-UnitStep[t-2])*Exp[-(t-2)/10]*Sin[t-2]; 
ic={y[0]==1,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((Heaviside(t - 2) - 1)*exp(1/5 - t/10)*sin(t - 2) + 3*y(t) + Derivative(y(t), 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)
 

Timed Out