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: y^iv + 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