MATH 238_Project 3_11.28.23
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University of Alabama *
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Course
238
Subject
Mathematics
Date
Jan 9, 2024
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docx
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4
Uploaded by ChancellorGalaxyAlpaca15
Olivia Campbell
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MATH 238 – Project 3
11/28/23
Part 1
: A mass-spring motion is governed by the ordinary differential equation
where m is the mass, b is the damping constant, k is the spring constant, and F(t) is the external force. We consider the initial conditions x(0) = 1 and x (0) = 0. Assume the following numerical ′
values for this part of the project: m = 1, k = 1/5, b = 1/5, and F(t) = cos γt.
(a)
Read section 4.10. Explain what the resonance frequency is, and then compute the resonance frequency for this mass-spring system.
The resonance frequency of a mass-spring system is the natural frequency where
a medium vibrates at the highest amplitude. The resonance frequency of this mass spring system is f=0.65 rad/s at ?
=4.2
(b)
The ODE45-solver can be used to obtain the solution curves in MATLAB. Use the provided MATLAB code to plot the solutions and estimate the amplitude A of the steady response for γ = 0.2, 0.42, 0.6, and 0.8.
?
= 0. 2 ?
=6.1
?
= 0. 4 ?
=11.2
?
= 0. 6 ?
= 5
?
= 0. 8 ?
= 2.1
(c) Plot the graph of A versus γ. For what frequency γ is the amplitude the greatest? Is it equal to that you obtained in (a)?
Amplitude is greatest at ?
=4.3, where A=11.5. This is very close to what I solved for in part a.
Part 2
: (Variable Coefficient Equations) A linear second-order differential equation that can be expressed in the form
where a, b, and c are constants, is called a Cauchy-Euler equation. We consider the homogeneous case (f(t) = 0) in this part. Note that the solutions should have the form because then (a)
Show that substituting y = t^r into the Cauchy-Euler equation yields the characteristic equation ar2 + (b − a)r + c = 0.
y=t^r
y’=rt^(r-1)= (ry)/t
y’’=(r^2)(r-1)t^(r-2)=(r(r-1)y)/t^2
Plug into equation:
at^(2)*(r(r-1)t^(r-2))+bt((r)*t^(r-1))+c(t^r)=0
ar(r-1)(t^r)+brt^r+ct^r=0
(ar(r-1)+br+c)t^r=0 t^r=0
a(r^2)-1ar+br+c=0
a(r^2)+(b-a)r+c=0
(b)
Find the roots of the characteristic equations of t^2y
+ 7ty − 7y = 0, then fnd the ′′
′
general solution to this equation.
a=1, b=7, c=-7
r^2+(7-1)r-7=0
r^2+6r-7=0
Real roots: r=1,-7
General solution: y(t)=c1t+c2/t^7
(c) Find the general solution to
In MATLAB, the dsolve command can be used to solve the ODE’s. For example, if we want to solve the equation 5y + 6y − 8y = e −2t , we can use the following commands: ′′
′
>> syms y(t) >> ode = 5
∗
diff(y, t, 2) + 6
∗
diff(y, t, 1) − 8 ∗
y == exp(−2t) >> sol(t) = dsolve(ode) Explanation: (1). The syms command defnes y as a function of the variable t. (2). diff(y, t, 2) is the 2nd derivative of y, and diff(y, t, 1) is the 1st derivative. So, the second command defnes the differential equation 5y + 6y −8y = e −2t , and we save this ′′
′
equation as ”ode”. (3). The 3rd command uses dsolve(ode) to solve this differential equation, then save the solution as ”sol(t)”. Based on the example above, use the dsolve command to solve the two equations in part (b) and (c). Compare the results with your answers.
t^2(y’’)+6ty’+4y=0
a=1, b=6, c=4
r^2=(6-1)r+4=0
Real roots: r=-1,-4
General solution: y(t)=c1t^-1+c2t^-4
For equation b I got the same solution on matlab.
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For equation c, I also got the same solution