72.16.25 problem 26

Internal problem ID [14921]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.1 page 399
Problem number : 26
Date solved : Tuesday, January 28, 2025 at 07:20:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=2 \,{\mathrm e}^{-2 t} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)+4*y(t)=2*exp(-2*t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 25

DSolve[{D[y[t],{t,2}]+4*y[t]==2*Exp[-2*t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

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