## Find the Inverse Laplace Transforms

Find the inverse Laplace transform of: 3s+5s(s2+9) Workings: 3s+5s(s2+9) =3ss(s2+9+5s(s2+9) =3s2+9+5s1s2+9 =sin(3t)+5s1s2+9 Now I’m not to sure on what to do. Any help will be appreciated. Answer The convolution theorem is, found here, L−1{f(s)g(s)}=∫t0f(t−u)g(u)du In the case here f(s)=1/s which is the transform of 1 and g(s) being the transform of sin. Now L−1{1s(s2+a2)}=1a∫t0(1)sin(au)du=−1a[cos(au)a]t0=−1a2(cos(at)−1) With … Read more

## Omega limit set vs. stable manifold of a point?

What is relationship between omega limit of a point and its stable manifold ? On mathoverflow I would the exact same question that hasn’t an answer, but only a comment as explanation saying a quick answer is that if x is in the omega limit set of ξ, then ξ is in the stable manifold … Read more

## Find all line equations that are tangent to x3−xx^3 – x and pass through (−2,2)(-2,2)

So I have the equation: f(x)=x3−x So we know that the slope of the curve for some x is given by: f′(x)=3×2−1 And need to find equations of lines that are tangent to that curve, and also pass through the point (−2,2). I have seen a similar question involving a circle equation, but couldn’t get … Read more

## Why $x^2$ may not be a solution to a general second order homogenous differential equations

I’m relatively new to differential equations and I recently read that if the functions $p(x)$ and $q(x)$ are continuous at $x=0$ then it is never possible for $x^2$ to be a solution to the equation $y”+p(x)y’+q(x)y = 0$ Given that this is still relatively elementary work as I’ve only just started looking at second order … Read more