14.11.9 problem 9

Internal problem ID [2602]
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 : 9
Date solved : Sunday, March 30, 2025 at 12:11:17 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=2 \cos \left (t \right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+5*y(t) = 2*cos(t)^2; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (85 c_2 \,{\mathrm e}^{t}-20\right ) \sin \left (2 t \right )}{85}+{\mathrm e}^{t} \cos \left (2 t \right ) c_1 +\frac {\cos \left (2 t \right )}{17}+\frac {1}{5} \]

✓ Mathematica. Time used: 0.388 (sec). Leaf size: 39

ode=D[y[t],{t,2}]-2*D[y[t],t]+5*y[t]==2*Cos[t]^2; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \left (\frac {1}{17}+c_2 e^t\right ) \cos (2 t)+\left (-\frac {4}{17}+c_1 e^t\right ) \sin (2 t)+\frac {1}{5} \]

✓ Sympy. Time used: 2.279 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 2*cos(t)**2 - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )}\right ) e^{t} - \frac {4 \sin {\left (2 t \right )}}{17} + \frac {\cos {\left (2 t \right )}}{17} + \frac {1}{5} \]