Mat 303 Calculus Iv Homework 7 Worksheet With Answers - State University Of New York At Stony Brook - 2013

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MAT 303 Spring 2013
Calculus IV with Applications
Homework #7 Solutions
Problems
• Section 3.3: 4, 18, 22, 24, 34, 40
• Section 3.4: 4, 12abc, 16, 18, 22. Omit the graphing part on problems 16 and 18.
3.3.4. Find the general solution to the differential equation 2y
7y
3y
0.
2
Solution: We determine that the characteristic equation for this linear polynomial is 2r
7r
3
0, which factors as 2r
1 r
3
0. Thus, the roots are r
1/2 and r
3, so
x/2
3x
the general solution is y
c
e
c
e
.
1
2
4
3.3.18. Find the general solution to the differential equation y
16y.
4
4
Solution: Writing the DE as y
16y
0, we see that its characteristic equation is r
16
0. Since this is a difference of squares, it factors as
2
2
2
r
4 r
4
r
2 r
2 r
4
0.
2
Therefore, it has the roots r
2 and r
2 from the linear factors, and the r
4 factor
has pure imaginary roots r
4
2i. From the real roots, we have the solutions
2x
2x
e
and e
, while we get the trigonometric functions cos 2x and sin 2x from the pure
imaginary roots. Thus, the general solution is
2x
2x
y
c
e
c
e
c
cos 2x
c
sin 2x.
2
3
1
4
3.3.22. Solve the IVP 9y
6y
4y
0, y 0
3, y 0
4.
Solution: We first find the general solution to the homogeneous linear DE. Since its char-
2
acteristic equation is 9r
6r
4
0, which has roots
2
6
6
4 4 9
1
3
1
1
r
i,
2 9
3
3
3
the general solution is the linear combination
x
x
x/3
x/3
y
c
e
cos
c
e
sin
.
1
2
3
3
1

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