We begin the
question by finding the general solution of this second order differential equation. In order to
approach this question we need to find the characteristic equation first:
The characteristic equation is found as follows:
0
Now we apply basic factorization:
0
Now we determine the roots by equating each term to zero:
From the
above roots we can now find the general solution:
e^(-x)
where:
are constants.
Since we have conditions, y(0) = 2 and y'(0) = 1, we can find the particular solution
and solve for the above constants.
Let's begin with the first constraint:
y(0) = 2
, e^0 = 1
= C_1 + C_2
Now we use the second constraint y'(0)=1. But first
we must find y'(x)
(we know from above e^0 =1)
(equation 2)
We have two unknowns and two equations. We can now add both
equations:
C_1
Now we can find C_2 from equation
1:
Now we have our
constants our particular equation is:
e^(-3x) + (5/2)e^-x
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