Ordinary Differential Equations
MATH 308 - 523
A. Bonito
March 3rd
Spring 2015
Last Name:
First Name:
Quiz 3
• 5 minute individual quiz;
• Answer the questions in the space provided. If you run out of space, continue
onto the back of the page. Additional space is provided at the end;
• Show and explain all work;
• Underline the answer of each steps;
• The use of books, personal notes, calculator, cellphone, laptop, and communication with others is forbidden;
• By taking this quiz, you agree to follow the university’s code of academic integrity.
Exercise 1
100%
Find the general solution to the following ODE
y ′′ + y = x sin(x)
and graph the evolution of y(x) for large values of x > 0.
Ordinary Differential Equations
MATH 308 - 523
A. Bonito
February 17
Spring 2015
Quiz 2: solutions
Exercise 1
100%
We first consider the homogeneous equation by solving the characteristic equation
λ2 + 1 = 0.
This is λ = ±i. Therefore two linearly independent solutions of the homogeneous equation are
given by
y1 (x) = Re(eix ) = cos(x),
y2 (x) = Im(eix ) = sin(x).
We now guess a particular solution of the form
yp (x) = Im(zp (x)),
where
zp (x) = wp (x)eix .
Pluging zp (x) into the ODE we get
wp′′ + 2iwp′ = x,
(1)
leading to the educated guess for wp (x)
wp (x) = Ax2 + Bx
for some constants A and B. Pluging wp (x) in (1) yields
i
A=− ,
4
B=
1
.
4
Therefore,
yp (x) = Im(zp (x)) = Im
=
i
1
− x2 + x (cos(x) + i sin(x))
4
4
x2
1
x sin(x) −
cos(x).
4
4
and
1
x2
y(x) = C1 cos(x) + C2 sin(x) + x sin(x) −
cos(x),
4
4
for some constants C1 and C2 .
For large values of x the solution y(x) looks like
y(x) ≈ −
The graph is provided in Fig. 1.
x2
cos(x).
4
2
Figure 1: graph of y(x) = − x4 cos(x).