72.17.11 problem 11

Internal problem ID [14947]
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 4.2 page 412
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 07:23:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+8 y&=\cos \left (t \right ) \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: 25

dsolve([diff(y(t),t$2)+6*diff(y(t),t)+8*y(t)=cos(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {2 \,{\mathrm e}^{-4 t}}{17}+\frac {7 \cos \left (t \right )}{85}+\frac {6 \sin \left (t \right )}{85}-\frac {{\mathrm e}^{-2 t}}{5} \]

✓ Solution by Mathematica

Time used: 2.029 (sec). Leaf size: 202

DSolve[{D[y[t],{t,2}]+5*D[y[t],t]+8*y[t]==Cos[t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-5 t/2} \left (-\sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^0\frac {2 e^{\frac {5 K[1]}{2}} \cos (K[1]) \cos \left (\frac {1}{2} \sqrt {7} K[1]\right )}{\sqrt {7}}dK[1]+\sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^t\frac {2 e^{\frac {5 K[1]}{2}} \cos (K[1]) \cos \left (\frac {1}{2} \sqrt {7} K[1]\right )}{\sqrt {7}}dK[1]+\cos \left (\frac {\sqrt {7} t}{2}\right ) \left (\int _1^t-\frac {2 e^{\frac {5 K[2]}{2}} \cos (K[2]) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )}{\sqrt {7}}dK[2]-\int _1^0-\frac {2 e^{\frac {5 K[2]}{2}} \cos (K[2]) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )}{\sqrt {7}}dK[2]\right )\right ) \]