Download MATH 54 − HINTS TO HOMEWORK 11 Here are a couple of hints to

MATH 54 − HINTS TO HOMEWORK 11
PEYAM TABRIZIAN
Here are a couple of hints to Homework 11. Make sure to attempt the
problems before you check out those hints. Enjoy!
S ECTION 4.6: VARIATION OF PARAMETERS
The easiest way to do the problems in this section is to look at my differential equations handout!
The formula is:
Let y1 (t) and y2 (t) be the solutions to the homogeneous equation, and
suppose yp (t) = v1 (t)y1 (t) + v2 (t)y2 (t). Let:
y
(t)
y
(t)
1
2
f (t) = 0
W
y1 (t) y20 (t)
And solve:
0 0
v1 (t)
f
=
W (t) 0
v2 (t)
f (t)
where f (t) is the inhomogeneous term.
S ECTION 6.1: BASIC THEORY OF LINEAR DIFFERENTIAL EQUATIONS
6.1.2, 6.1.4. First of all, make sure that the coefficient of y 000 is equal to 1.
Then look at the domain of each term, including the inhomogeneous term
(more precisely, the part of the domain which contains the initial condition).
Then the answer is just the intersection of the domains you found!
6.1.10. Use the Wronskian with x = 0
6.1.12. cos2 (x) − sin2 (x) = cos(2x)
6.1.17. Verify that the three functions solve the differential equations, then
show they’re linearly independent (by using the Wronskian at x = 1)
Date: Tuesday, April 14th, 2015.
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PEYAM TABRIZIAN
6.1.21. Use the Wronskian to check that the functions are linearly independent (try to simplify first). Then the general solution is y(t) = Ax +
Bx ln(x) + Cx (ln(x))2 + ln(x), and then plug in the initial condition.
6.1.34. It’s third-order because it involves y 000 . In fact the coefficient of y 000
is 1, by expanding the determinant along the fourth row. What happens
when you plug in y(t) = f1 (t), is the determinant in fact 0 or not? Is f1 (t)
a solution or not?
S ECTION 6.2: H OMOGENEOUS LINEAR EQUATIONS WITH CONSTANT
COEFFICIENTS
6.2.3, 6.2.9, 6.2.19, 6.2.21. The following fact might be useful:
Rational roots theorem: If a polynomial p has a zero of the form r = ab ,
then a divides the constant term of p and b divides the leading coefficient of
p.
This helps you ‘guess’ a zero of p. Then use long division to factor out p.
6.2.14. Since sin(3x) is a solution, this means that one of the roots of the
auxiliary equation is r = ±3i, which tells you that the auxiliary equation
has the form (r2 + 9)× something. Use long division to figure out what that
something is, by dividing r4 + 2r3 + 10r2 + 18r + 9 by r2 + 9.
6.2.17. The reason this is written out in such a weird way is because the
auxiliary polynomial is easy to figure out! Here, the auxiliary polynomial
is
(r + 4)(r − 3)(r + 2)3 (r2 + 4r + 5)2 r5 = 0.
S ECTION 9.1: I NTRODUCTION
k
9.1.7. Let z = y 0 , then z 0 = y 00 = − mb y 0 − m
y = − mb z −
(
y0 = z
k
y − mb z
z0 = − m
k
y,
m
then we get:
MATH 54 − HINTS TO HOMEWORK 11
3
9.1.11, 9.1.12. For 9.1.11, let:

x1 = x



x = x 0 = x 0
2
1

x
=
y
3



x4 = y 0 = x03
Now calculate x0i and put them in a system, using the above relations. In
particular x02 = x00 = −3x − 3y = −3x1 − 2x3 .