14.11.12 problem 12

Internal problem ID [2605]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.5. Method of judicious guessing. Excercises page 164
Problem number : 12
Date solved : Monday, January 27, 2025 at 06:02:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+4 y&=t^{2}+\left (2 t +3\right ) \left (1+\cos \left (t \right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 58

dsolve(diff(y(t),t$2)+diff(y(t),t)+4*y(t)=t^2+(2*t+3)*(1+cos(t)),y(t), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {15}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {15}\, t}{2}\right ) c_1 +\frac {17}{32}+\frac {\left (6 t +5\right ) \cos \left (t \right )}{10}+\frac {\left (2 t +5\right ) \sin \left (t \right )}{10}+\frac {t^{2}}{4}+\frac {3 t}{8} \]

✓ Solution by Mathematica

Time used: 7.218 (sec). Leaf size: 87

DSolve[D[y[t],{t,2}]+D[y[t],t]+4*y[t]==t^2+(2*t+3)*(1+Cos[t]),y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {t^2}{4}+\frac {3 t}{8}+\frac {1}{5} t \sin (t)+\frac {\sin (t)}{2}+\frac {1}{10} (6 t+5) \cos (t)+c_2 e^{-t/2} \cos \left (\frac {\sqrt {15} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {15} t}{2}\right )+\frac {17}{32} \]