Dynamic Engineering Systems and Mathlab assignment
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homework_15.pdf
Drexel University, College of Engineering 2015-2016 Academic Year
Drexel University Office of the Dean of the College of Engineering
ENGR 232 – Dynamic Engineering Systems
Homework 1 – Week 1
The following problems are assigned from the main course textbook: Differential Equations and Linear Algebra (Farlow and Hall); solutions will be posted in a couple of days. The material tested here will appear on Week 2 Quiz 1.
Problem 1.
Fill in the following table, specifying the dependent variable, the independent variable(s), the order of the differential equation and whether it is linear or non-linear and explain why. If the system is non- linear, place a box around the term(s) making it non-linear. All equations are in terms of a function of an independent variable.
SYSTEM ORDER?
Dependent Variable
Independent Variable(s)
LINEAR OR NON-LINEAR? Circle any term(s) that make it
non-linear.
x2y'' + xy' + y = cos (x)
y' + ey = x
2
2
dt yd
= -t2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
+ et
)ln(3 3
xt dt dxx
dt xd
=+
)sin(2 2
ktE k i
dt diR
dt idL =++
( ) ktyty dt
yd 125)/1(34 4
=−−
k>0
)sin()sin( t=−′+ θθθ
txtxt ′′−=−′ 5)(sin22
Drexel University, College of Engineering 2015-2016 Academic Year
( dt dy
)2 + dt dy
= R cos(t)
)2()1( −−−= tt dt da
)sin(2 tty dt dyy =−
y(t)𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
= -ω t
𝑡𝑡 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
= y(x-t)
)ln(3 3
tt dt dx
dt xd
=+
3
3
dt xd
+ kRt = sin (t) * exp(t)
Problem 2
Solve the IVP numerically
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
= f(t, y) = t - y.
Let t0 = 0, y(0) = 1
Additionally, choose t to be in increment of 0.5 until you reach t = 2.5. Show your result in tabular form using the format below:
t Euler (yn) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑡𝑡
0 0 1 0.5 2 1 3 1.5 4 2 5 2.5
Drexel University, College of Engineering 2015-2016 Academic Year
Problem 3
Solve the IVP numerically
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
= f(t, y) = 4 - 2t + 2.5y
Let t0 = 0, y(0) = 1
Additionally, choose t to be in increment of 0.1 until you reach t = 0.4. Show your result in tabular form using the format below:
t Euler (yn) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑡𝑡
0 0 1 0.1 2 0.2 3 0.3 4 0.4
Drexel University, College of Engineering 2015-2016 Academic Year
Problem 4
Given the direction field for 𝒙𝒙′ = −𝟏𝟏 𝟒𝟒 𝒙𝒙(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟐𝟐) showing x vs. t
a) How many equilibrium solutions for this differential equation? List in the box:
b) Estimate the time for the initial condition x(0) = -1 to reach its equilibrium solution. Draw the curve on the direction field and show the time, fill in the values below
c) Draw the solution curve for x(0) = 1 on the direction field.
Drexel University, College of Engineering 2015-2016 Academic Year
