14.11.10 problem 10

Internal problem ID [2603]
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 : 10
Date solved : Tuesday, March 04, 2025 at 02:30:10 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 30

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

\[ y = \frac {{\mathrm e}^{t} \left (\left (4 c_1 +1\right ) \cos \left (2 t \right )+1+\left (t +4 c_2 \right ) \sin \left (2 t \right )\right )}{4} \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 36

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

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

✓ Sympy. Time used: 0.820 (sec). Leaf size: 26

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

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