90.15.11 problem 12

Internal problem ID [25258]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 3. Second Order Constant Coefficient Linear Differential Equations. Exercises at page 251
Problem number : 12
Date solved : Thursday, October 02, 2025 at 11:59:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+2*y(t) = exp(t)*cos(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (\sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 \right ) {\mathrm e}^{-t}+\frac {{\mathrm e}^{t} \left (\cos \left (t \right )+\sin \left (t \right )\right )}{8} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 41

ode=D[y[t],{t,2}]+2*D[y[t],{t,1}]+2*y[t]==Exp[t]*Cos[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{8} e^{-t} \left (\left (e^{2 t}+8 c_2\right ) \cos (t)+\left (e^{2 t}+8 c_1\right ) \sin (t)\right ) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*y(t) - exp(t)*cos(t) + 3*Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} \sin {\left (\frac {\sqrt {6} t}{3} \right )} + C_{2} \cos {\left (\frac {\sqrt {6} t}{3} \right )} + \frac {3 e^{t} \sin {\left (t \right )}}{20} + \frac {e^{t} \cos {\left (t \right )}}{20} \]