Find the general solution for the system:
dx/dt = 2x + y + 2t
dy/dt = 2x + 3y + t
dy/dt = 2x + 3y + t
Alert readers will see that the homogenous form for the above has already been solved (July 1st), e.g. http://brane-space.blogspot.com/2013/07/solution-to-problem-1-of-june-30-set.html
So we know the homogenous part of the soln is:
So we know the homogenous part of the soln is:
x = exp (4t) + exp (t)
and
y = 2 exp(4t) - exp (t)
Now, rewrite the forcing terms for the system as F(t) =
[2] t
[1]
[1]
And try a particular soln. of the system possessing the same form so that: Xf = D1 t
Where D1 =
[a2]
[b2]
[b2]
Then we proceed by setting: X’f = C (Xf) + C1 t
C is the original coefficient matrix, e.g.
[2.....1]
[2 ....3]
[2 ....3]
And C1 is the (2, 1) column matrix, with a factor t. Then we will need to solve (by undetermined coefficients):
[a2]
[b2] = C [ D1 t]
Writing all this out one should obtain:
2a2 + b2 + 2 = 0
2a2 + 3b2 + 1 = 0
For which we obtain: a2 = -5/4 and b2 = ½
The forcing components of the solution are then: Xf1 = -5 t/4 and Xf2 = t/2
The general solution is then:
x = c1exp (4t) + c2exp (t) – 5t/4
y = 2c1 exp(4t) - c2exp (t) + t/2
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