72.17.15 problem 15

Internal problem ID [14951]
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 : 15
Date solved : Tuesday, January 28, 2025 at 07:24:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+y&=\cos \left (3 t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 39

dsolve(diff(y(t),t$2)+3*diff(y(t),t)+y(t)=cos(3*t),y(t), singsol=all)
 

\[ y = {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}} c_{2} +{\mathrm e}^{-\frac {\left (\sqrt {5}+3\right ) t}{2}} c_{1} -\frac {8 \cos \left (3 t \right )}{145}+\frac {9 \sin \left (3 t \right )}{145} \]

✓ Solution by Mathematica

Time used: 0.279 (sec). Leaf size: 112

DSolve[D[y[t],{t,2}]+3*D[y[t],t]+y[t]==Cos[3*t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-\frac {1}{2} \left (3+\sqrt {5}\right ) t} \left (\int _1^t-\frac {e^{\frac {1}{2} \left (3+\sqrt {5}\right ) K[1]} \cos (3 K[1])}{\sqrt {5}}dK[1]+e^{\sqrt {5} t} \int _1^t\frac {e^{-\frac {1}{2} \left (-3+\sqrt {5}\right ) K[2]} \cos (3 K[2])}{\sqrt {5}}dK[2]+c_2 e^{\sqrt {5} t}+c_1\right ) \]