14.11.4 problem 4

Internal problem ID [2597]
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 : 4
Date solved : Monday, January 27, 2025 at 06:01:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=t^{2}+t +1 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 37

dsolve(diff(y(t),t$2)+diff(y(t),t)+y(t)=1+t+t^2,y(t), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +t^{2}-t \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 54

DSolve[D[y[t],{t,2}]+D[y[t],t]+y[t]==1+t+t^2,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t/2} \left (e^{t/2} (t-1) t+c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]