Laplace Transform

Question

Use the Laplace method to solve the following IVP:



Answer 1

s²F(s) -3s + 16F(s) = 6/(s² + 9);

So, F(s) = 6/[(s² + 9)(s² + 16)] + 3s/(s²+ 16);

Then we decompose: 6/[(s² + 9)(s² + 16)] = (as + b)/(s² + 16) + (cs + d)/(s² + 9);

We get system: a + c = 0, b + d = 0, 16a + 9c = 0, 16b + 9d = 6;

I solve it using TI-84: a = 0, b = 6/7, c = 0, d = -6/7. So, F(s) = 6/7·1/(s² + 16) - 6/7·1/(s² + 9) + 3s/(s² + 16)

Now Inverse Laplace transformgive us: x(t) = 6/7·1/4sin4t - 6/7·1/3sin3t + 3·cos4t;

x(t) = 3/14sin4t - 2/7sin3t + 3cos4t