72.16.24 problem 25

Internal problem ID [14920]
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 : 25
Date solved : Tuesday, January 28, 2025 at 07:20:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+9 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: 0.019 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)+9*y(t)=exp(-t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\sin \left (3 t \right )}{30}-\frac {\cos \left (3 t \right )}{10}+\frac {{\mathrm e}^{-t}}{10} \]

✓ Solution by Mathematica

Time used: 0.120 (sec). Leaf size: 113

DSolve[{D[y[t],{t,2}]+9*y[t]==Exp[-t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\sin (3 t) \int _1^0\frac {1}{3} e^{-K[2]} \cos (3 K[2])dK[2]+\sin (3 t) \int _1^t\frac {1}{3} e^{-K[2]} \cos (3 K[2])dK[2]+\cos (3 t) \left (\int _1^t-\frac {1}{3} e^{-K[1]} \sin (3 K[1])dK[1]-\int _1^0-\frac {1}{3} e^{-K[1]} \sin (3 K[1])dK[1]\right ) \]