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Transcript
Name________________________
Student I.D.___________________
Math 2250−1
Quiz 7
October 21, 2011
SOLUTIONS
1) Consider the differential equation for y x
y
5 y 6 y=0 .
1a) Find the general solution to this differential equation.
rx
rx
trying y x = e yields L y = e
of the characteristic polynomial
2
r
5 r
rx
6
so in order for e
(6 points)
to be a solution, r must be a root
r2 5 r 6 = r 3 r 2 .
Since the roots are r = 3, 2 the general solution is
yH x = c1 e 3 x c2 e 2 x .
1b) What is the dimension of the solution space above?
3x
Since there are two basis functions, y1 = e
2x
, y2 = e
(1 point)
for the solution space, the dimension is two.
1c) Use your work in (1a) to solve the initial value problem
y
5 y 6 y=0
y 0 = 1
y 0 =4 .
(3 points)
y x = c1 e
3x
c2 e
3x
y x = 3 c1 e
2x
2 c2 e
2x
at x = 0 :
1 = c1 c2
4 = 3 c1 2 c2 .
1
4
c1
c2
=
1
1
3
2
1
=
1
4
1
1
c1
3
2
c2
=
1
1
2
1
1
3
1
4
So
y x = 2 e
3x
e
2x
.
=
2
1
.