14.16.6 problem 20

Internal problem ID [2676]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.9, The method of Laplace transform. Excercises page 232
Problem number : 20
Date solved : Monday, January 27, 2025 at 06:06:16 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=t^{3} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.902 (sec). Leaf size: 43

dsolve([diff(y(t),t$2)+diff(y(t),t)+y(t)=t^3,y(0) = 2, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {4 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{3}+t^{3}-4 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )-3 t^{2}+6 \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 60

DSolve[{D[y[t],{t,2}]+D[y[t],t]+y[t]==t^3,{y[0]==2,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to t^3-3 t^2-\frac {4 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}-4 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+6 \]