need help in diffrential equation and laplace transformation?

Question

find the laplace transform of the solution of the equation
(D^2_5D'+4)y=12+9e^t+5Sin2t
which satisfies the conditions y(0)=1 y'(0)=-2
another question........
Factor the following diffrential operators
2D^2+9D-5

Answer 1

Assuming that your underscore above is supposed to be a subtract sign then I get the Laplace is

3/s-3/(s-1)^2+ 1/2*s/(s^2+4) -4/(s-1)+ 3/2/(s-4)

with solution

3-3*exp(t)*t+ 1/2*cos(2*t)- 4*exp(t)+ 3/2*exp(4*t)

You use the initial conditions when you apply the Laplace: for instance L(y')=s*L(y)-y(0)

Factoring a differential operator is like factoring a polynomial: 2a^2+9a-5 factors as (2a-1)(a+5) so 2D^2+9D-5 factors as (2D-I)(D+5I) where I is the identity funtcion: I(y)=y

Answer 2

(s^2 + 5s + 4)y(s) = 12/s + 9/(s - 1) + 5*4/(s^2 + 4)

Solve for y(s), then use partial fractions to apply the inverse transform.

12(s - 1)(s^2 + 4) + 9s(s^2 + 4) + 20s(s - 1)
y(s) = ----------------------------------------------------------
s(s - 1)(s^2 + 4)(s^2 + 5s + 4)

= A/s + B/(s - 1) + C/(s^2 + 4) + D/(s + 4) + E/(s + 1)

I would now use a calculator or Mathematica to find A, B, C, D, and E, and write down the time solution:

y(t) = A + Be^t + C/2*sin(2t) + De^-4t + Ee^-t.

Then put in 0 for t and equate to the initial condition y(0) = 1.

Then differentiate, put in 0 for t and equate to the initial condition y'(0) = -2. The derivative is

y'(t) = Be^t + Ccos(2t) - 4De^-4t - Ee^-t.

The calculations are long, but not difficult. Hope this helps.