50.25.7 problem 4(c)

Internal problem ID [8179]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 Problesm for review and discovery. Section A, Drill exercises. Page 309
Problem number : 4(c)
Date solved : Wednesday, March 05, 2025 at 05:31:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method

✓ Maple. Time used: 1.395 (sec). Leaf size: 52

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+2*y(t) = t; 
dsolve(ode,y(t),method='laplace');
 

\[ y = -\frac {1}{4}+\frac {t}{2}+\frac {\left (7 \cos \left (\frac {\sqrt {7}\, t}{2}\right ) \left (1+4 y \left (0\right )\right )+\sin \left (\frac {\sqrt {7}\, t}{2}\right ) \sqrt {7}\, \left (8 y^{\prime }\left (0\right )+4 y \left (0\right )-3\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{28} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 56

ode=D[y[t],{t,2}]+D[y[t],t]+2*y[t]==t; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {t}{2}+c_2 e^{-t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )-\frac {1}{4} \]

✓ Sympy. Time used: 0.177 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + 2*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t}{2} + \left (C_{1} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {7} t}{2} \right )}\right ) e^{- \frac {t}{2}} - \frac {1}{4} \]