Solve the differential equation d^2y/dx^2+2 dy/dx-35y=0 Given that when x=0,y=5 and dy/dx=-1?

Question

Solve the differential equation d^2y/dx^2+2 dy/dx-35y=0 Given that when x=0,y=5 and dy/dx=-1?

Answer 1

find the roots:

D^2 + 2D - 35 = 0

(D + 7) (D - 5) = 0

the roots are -7 and 5.

thus

y = Ae^(-7x) + Be^(5x)

.. . .

when x = 0, y = 5 = A + B

y' = -7Ae^(-7x) + 5Be^(5x)

when x = 0 , y' = -1 = -7A + 5B

25 = 5A + 5B

1 = 7A - 5B

12A = 26

A = 13/6

B = 30/6 - 13/6 = 17/6

thus

y = 13/6 e^(-7x) + 17/6 e^(5x)