## Monday, August 12, 2013

### Solutions to 'Tricks for Solving DEs (Pt. 3)'

1) Find the general solution of: y”’ – 3y” + 7y’ – 5y = 0

The auxiliary equation is:

m3 - 3m2 + 7m - 5 = 0

Then the root m = 1 can be extracted using synthetic division so:

1….-3….+7…..-5 (1)
……1….-2……5
----------------------
1…..-2….5……0

And the resulting quadratic is:
m2 -2m  + 5 = 0

Solve using the quadratic formula to obtain: m = 1 +  2i

with a = 1 and b = 2.

The general solution can then be written:

y = c1 exp (x) + exp (x)[ c2 cos 2x + c3 sin 2x]

2)Find the general solution of: yiv + 18y” + 81 = 0

The auxiliary equation is: m4 + 18m2  + 81 = 0

It is fairly easy to factor this to get: (m2 + 9)2 = 0

And each (m2 + 9) is easily factored in turn to get:

(m + 3i)(m – 3i) = 0

i.e. m2 + 3mi – 3mi -9i2 =  m2 -9 (-1) = m2 + 9

The roots are doubly repeated and are: m= + 3i and + 3i

Hence a = 0 and b = 3 each time so the general solution can be written:

y = (c1 + c2x) cos 3x + (c3 + c4 x) sin 3x