# transforms

1. ### Finding solution to 2nd order using Laplace Transforms

Hi, I have this problem that states: Solve the following initial value problem: y'' - 25y = 50t, y(0) = 0, y'(0)=0 I get stuck at the end and I'm really confused about how to finish solving it so I was hoping someone on here could help me. Here's what I've done so far: Denote \mathcal{L}[y]...
2. ### Fourier transforms and Bessel functions

Hi I am asked to show that the Fourier transform of f(x)=\frac{1}{\sqrt{1-x^2}} is \tilde{f}(k)=\sqrt{\pi/2}J_0(-k) where J_0(x)=\frac{1}{\pi}\int_0^\pi e^{ix\cos\theta}d\theta and \tilde{f}(k)=\frac{1}{\sqrt{}2\pi}\int_{-\infty}^{\infty}e^{-ikx}\frac{1}{\sqrt{1-x^2}}dx I thought about...
3. ### Groebner for substitution/transform question.

I am trying to convert one quadratic polynomial p(x) to another one q(x) via: x'=m*x+n So I do the obvious and set p(x)-q(x')=0 x'=m*x+n In Axiom groebnerFactorize([p(x)-q(x'), x'=m*x+n],nozero,true) and the polynomials are set to have variables m,n. where nozero cuts out silly answers like m=0...
4. ### solve IVP using laplace transforms: please check my work in attached image

Original problem: y'' + 7y' + 10y = 3e^(-2t) - 6e^(-5t) and y(0) =0 and y'(0) = 0
5. ### Laplace Transforms - Check my work please

Hi, I'd appreciate it if someone could check my working for Question 4. (Sorry about the poor quality, let me know if you can't read it.) Question - imgur: the simple image sharer Working - imgur: the simple image sharer Thanks
6. ### Laplace Transforms

Hi, For Question 2, I know you can do them by partial intergration but I'm not sure how to show them using the method it asks. For Question 4, I genuinely have no idea. Any help would be appreciated. Thanks
7. ### Check my work - Fourier Transforms.

Question - imgur: the simple image sharer (Ignore after black line) Answers - imgur: the simple image sharer (Second image is first) I've yet to do part (c) but is my graph and transform correct?
8. ### Solving an I.V.P. with Laplace transforms.

y^{\prime\prime}+6y^\prime+9y=0 and y(0)=-1, y^\prime(0)=7 OK so I use two properties of Laplace Transforms; {\Lapl}{\mathcal{L}}(f^{\prime\prime})(s)=s^2 {\Lapl}{\mathcal{L}}(f)(s)-sf(0)-f^\prime(0) and {\Lapl}{\mathcal{L}}{(f^\prime)}(s) = s{\Lapl}{\mathcal{L}}(f)(s) - f(0) And I replace...
9. ### Partial Fractions and Laplace Transforms for Third-Order and Higher Equations

Hi, I'm having some difficulty with some Laplace transforms and was wondering if someone could give me a helping hand. I have some equations in the Laplace domain that I would like to put into the time domain. The equations I have are in the form of: X(s)/Y(s) = (As^3 + Bs^2 + Cs + D) / (Es^4...
10. ### What is the standard definition of transforms T_{1,-1} and D_2

What is the standard definitions of the following transforms? I got them from a practice test for the NY Regents. The test does not include those definitions in its reference sheet. T_{1,-1} D_2 I need this for the following: Triangle ABC has vertices A(5,1), B(1,4), and C(1,1). What are the...
11. ### Help with a couple differential equation problems.(mostly laplace transforms)

Please and thank you. Solve by undetermined coefficients y'' + 3y = -48x2 * e3x Find a linear operator that annihilates the given function, and verify 3 + ex * cos2x Inverse Laplace transforms of (s + 1)3/s4 0.9s/((s - 0.1)(s + 0.2)) Use the Laplace transform to solve the given initial...
12. ### Fourier transforms...?

we have these 2 functons: H(u)= A*e^[(-u^2)/2*sigma^2)] h(x)= sqrt(2pi)*sigma*A*e^[-2*(pi^2)*(sigma^2)*(x^2)] and also we have Fourier transforms. question: prove that H(u) is a transform of h(x), or h(x) is a transform of H(u).
13. ### Fourier transforms...?

we have these 2 functons: H(u)= A*e^[(-u^2)/2*sigma^2)] h(x)= sqrt(2pi)*sigma*A*e^[-2*(pi^2)*(sigma^2)*(x^2)] and also we have Fourier transforms. question: prove that H(u) is a transform of h(x), or h(x) is a transform of H(u).
14. ### Need help with simple unit step functions, (for Laplace transforms)?

Hi, here's my question... I am required to find the Laplace transform of the given function which is assumed to be zero outside the interval. t^2 (0<t<1) I have to first rewrite it in terms of unit step functions, and that's the step i dont understand? The solution manual writes it as...
15. ### Question on the applications of Laplace transforms

hi I need your help to resolve this question y''-y'-2y=18e^-t sin3t y(c)=0 , y'(c)=3 ?
16. ### Fourier transforms and their inverse

I'm having a little bit of trouble that I know should be trivial but for some reason I cannot get it. As a simple example consider f(x)&=&x. I want to perform a Fourier transform on f(x) and then perform an inverse Fourier transform on this transformed function to get back to the original...
17. ### Differential equations using Laplace transforms

y''+9y=r(t)=\left\{\begin{matrix} 8sin(t),t\in (0,\pi) \\ 0,t \in (\pi,+\infty) \end{matrix}\right. ,y(0)=0,y'(0)=4 How can I express r(t) using Dirac delta function or Heaviside step function? I know that Heaviside is the antiderivative of Dirac. H(x)=\left\{\begin{matrix} 0, x<0 \\...
18. ### Question about Laplace transforms

y''-4y'+5y=sin(t), t \in [0,+\infty), y(0)=0, y'(0)=0 s^{2}Y(s)-sy(0)-y'(0)-4(sY(s)-y(0))+5\frac{1}{s^{2}}=\frac{1}{s^{2}+1} <=> Y(s)(s^{2}-4s)=\frac{1}{s^{2}+1}-\frac{5}{s^{2}} <=> Y(s)(s^{2}-4s)=\frac{-4s^{2}-5}{s^{2}(s^{2}+1)} <=> Y(s)=\frac{-4s^{2}-5}{s^{2}(s^{2}+1)(s^{2}-4s)} <=>...
19. ### Engineering student

Hi all I am a mechatronics engineering student from Ireland. I am looking to improve my math skills in general, I am covering Laplace transforms anf Fourier transforms. I have a reasonable understanding of both but need help with some aspects. Hope you can help.(Happy)
20. ### Inverse Laplace transforms

Hi guys, I'm really stuck on a problem and would really appreciate some help! I'm trying to find the inverse Laplace transform of a function with complex numbers in the form : 4exp(-s)/(s+5.5-i4.97) Thank you for any help