Can't solve this ODE using Laplace transforms!

Question

Can't seem to solve this. I get an expression as follows

$d^2x/dt^2 + x = 2t$

$X(s) = (2+5s^3)/s^2(s^2+1)$

But solving this using partial fractions doesn't get me the right answer. I would really appreciate some guidance. Thanks in advance!

Answer 1

Let's see if my decades of experience teaching DE can guess your mistake using almost no context:

When $s^2$ appears in the denominator, a lot of students make the mistake of treating it as an irreducible quadratic. They put

$$\frac{As+B}{s^2}$$

in the Ansatz. But this is wrong. It's really the linear factor $s$ to the 2nd power. So they should have written

$$\frac{A}{s} + \frac{B}{s^2}.$$

How good was my guess?