72.16.29 problem 30

Internal problem ID [14925]
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 : 30
Date solved : Tuesday, January 28, 2025 at 07:20:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y&=-{\mathrm e}^{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: 0.033 (sec). Leaf size: 28

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

\[ y = \frac {\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{6}+\frac {\cos \left (\sqrt {2}\, t \right )}{3}-\frac {{\mathrm e}^{t}}{3} \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 39

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

\[ y(t)\to \frac {1}{6} \left (-2 e^t+\sqrt {2} \sin \left (\sqrt {2} t\right )+2 \cos \left (\sqrt {2} t\right )\right ) \]