72.16.39 problem 41

Internal problem ID [14935]
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 : 41
Date solved : Tuesday, January 28, 2025 at 07:21:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=t +{\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: 0.017 (sec). Leaf size: 26

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

\[ y = -\frac {\sin \left (2 t \right )}{40}-\frac {\cos \left (2 t \right )}{5}+\frac {t}{4}+\frac {{\mathrm e}^{-t}}{5} \]

✓ Solution by Mathematica

Time used: 0.393 (sec). Leaf size: 147

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

\[ y(t)\to -\sin (2 t) \int _1^0\frac {1}{2} e^{-K[2]} \cos (2 K[2]) \left (e^{K[2]} K[2]+1\right )dK[2]+\sin (2 t) \int _1^t\frac {1}{2} e^{-K[2]} \cos (2 K[2]) \left (e^{K[2]} K[2]+1\right )dK[2]+\cos (2 t) \left (\int _1^t-e^{-K[1]} \cos (K[1]) \left (e^{K[1]} K[1]+1\right ) \sin (K[1])dK[1]-\int _1^0-e^{-K[1]} \cos (K[1]) \left (e^{K[1]} K[1]+1\right ) \sin (K[1])dK[1]\right ) \]