Use Laplace Transforms to solve these IVP (partial fractions):
1. x''+4'+8x=e^-t x(0)=x'(0)=0
I don't know how to work with the imaginary roots.
2. x''''-x=0 x(0)=1 x'(0)=x''(0)=x'''(0)=0
3. x''''+13x''+36x=0 x(0)=x''(0)=0 x'(0)=2 x'''(0)=-13
Apply the convolution theorem to find the inverse Laplace transforms for:
4. F(s) = 1/(s(s^2+4))
5. F(s) = 1/(s(s^2+4s+5))
Apply the convolution theorem to derive the indicated solution x(t) of the given DE w/ initial conditions x(0)=x'(0) = 0
6.x''+4x'+13x=f(t);
x(t)=1/3 int[0,t] f(t-tau)e^-2(tau)*sin(3*tau) dtau
Solve the IVP
7. mx''+cx'+kx=f(t);
x(0) = x'(0)=0
m=1
k=4
c=5
f(t)=1 if 0=<t=<2
f(t)=0 if t>=2
Help on any is useful.
Apply the convolution theorem to derive the indicated solution x(t) of the given DE w/ initial conditions x(0)=x'(0) = 0
x''+4x'+13x=f(t);
x(t)=1/3 int[0,t] f(t-tau)e^-2(tau)*sin(3*tau) dtau
taking the Laplace transform we get
by the convolution theorem we get
There are few ways the most common is to memorize some common formulas and use them back wards. If you have taken a course in complex variables I will show you how to evaluate the contour integral
so we want to use this in reverse so
by using the above formula backwards we get
Thanks again, that makes perfect sense.
I'm trying to decide between PDEs and Complex Variables. I think PDEs would be more useful in my field (atmospheric science), but I don't really know what the Complex Variables course is about.
So, for the first problem, I have it here:
With Roots: 2+-2i, what happens now?
I can tell there will be an e^-2t and a cos(2t) sin(2t), but im hazy for the steps to get to the final answer