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

First-order linear ordinary differential equation with piecewise constant source term

Find a continuous solution satisfying the DE: dydx+2y=f(x) f(x)={1,0≤x≤30,x>3.y(0)=0 I don’t get this problem at all. Can anyone explain what the above means for starters? Answer solve the equation in two pieces and then match it at the boundary x=1. the two problems ares y′+2y=1,y(0)=0 has the solution y=12(1−e−2x),x≤3. now solve y′+2y=0,y(3)=12(1−e−6). the solution is … Read more

Differential equation y″y” \cdot y^3 = 1

I use these substitutions y’=p(y) and y” = p’ \cdot p to solve the equation, thus I have the consequence of the solution’s steps: p’py^3 = 1 \implies p’p = \frac{1}{y^3} \implies \frac {dp}{dy} p = \frac {1}{y^3} \implies \int p dp = \int \frac{1}{y^3} dy \implies \\ p = \sqrt{C_1 – \frac {1}{y^2}} Then … Read more

Hints on solving y′=y3x−y2y’=\frac{y}{3x-y^2}

y′=y3x−y2 My attempt: dydx=y3x−y2 dy⋅(3x−y2)=dx⋅y dy⋅3x−dy⋅y2=dx⋅y Any direction? I need hints please not a full answer Answer I would write it as dxdy=3x−y2y=3xy−y Just maybe AttributionSource : Link , Question Author : 3SAT , Answer Author : Chinny84