Homework Statement
While t<1 , f(t) =0
While t>= 1, f(t) = t^2-2t+2
Find the LaPlace transform of the given function.
The attempt at a solution
The problem can be re-written as: u1(t)(t^2-2t+2)
If it helps, this in turn can be re-written as: u1(t)((t-1)^2+1). I'm not sure it does help...
Homework Statement
Find the Laplace transform of the given function, a and b are real constants.
Hint: cos(bt)=(e^ibt+e^-ibt)/2 and sin(bt)=(e^it-e^ibt)/2i.
f(t)=e^(at)*sin(bt)
The attempt at a solution
The LaPlace transform of e^at is 1/(s-a).
The LaPlace transform of sin(bt) is...
When determining a particular solution to a differential equation, one of the necessary steps is to ask the "homogeneous question" aka Does any term in yp solve the homogeneous equation for this problem. When it does, I know that it is necessary to multiple by t.
My question is, do I multiply...
Homework Statement
A mass weighing 2 lbs stretches a spring 6 inches. If the mass is then pulled down an additional 3 inches and then released, and there is no damping, determine the position u of the mass at ay time T according to the form Acos(wT-phi).
The attempt at a solution
The...
Homework Statement
Find the general solution for:
y''+2y'+5y=3sin2t
The attempt at a solution
y''+2y'+5y=3sin2t
First step is to find the general solution to the homogenous equation, so skipping 2 steps (letting y=e^rt and dividing)
R^2+2r+5
(-2+/- sqroot(4-4*5))/2
=-1 +/- 2i...
Homework Statement
The half life of carbon 14 is 5730 years long. Assume that dQ/dt = -rQ. Determine the constant rate r.
The attempt at a solution
dQ/dt = -rQ (Given)
dQ/Q = -rdt (algebra)
lnQ = -rt (integrate)
-lnQ/t = r (algebra)
How do I work in...
Homework Statement
Find the general solution for y' +3y = t + e^(-2t) for y.
The Attempt at a Solution
At first I thought that since the equation was already separated, I could simply integrate both sides and get a solution easily:
That results in 1.5 y^2 + y = .5t^2 - .5e^(-2t)
(Unless I...
Homework Statement
Find the plane that contains the line x=3+2T, Y=t, Z=8-t and is parallel to the plane 2x + 4y +8z =17.
The attempt at a solution
It seemed to me that since the plane is parallel to the plane 2x+4y+8z=17, it must be of the form 2x+4y+8z+D=0 so that it would be parallel...
Homework Statement
Find a nonzero vector orthogonal to the plane through points P (0, -2, 0) Q (4, 1, -2) and R (5,3,1) and find the area of the triangle formed by PQR.
The attempt at a solution
To be honest, I am not entirely sure how to do this problem. I've looked through my textbook...
Homework Statement
Show that y = x-(x^-1) is a solution for the differential equations:
xy' +y = 2x
The attempt at a solution
Frankly, I haven't a clue how to start.
If it weren't for the x in the xy' term this would be easy, since I could just integrate; unfortunately, I don't know...
Homework Statement
Integrate the square root of (x^2 -1) over the interval 0-1.
The attempt at a solution
First off, I know this is a quarter of a circle, but I'm not supposed to solve it that way.
Now then:
x = sec theta
dx= sec tan theta.
square root (x^2-1) dx
becomes...
Homework Statement
Integrate: (x-1) / (X^2 - 4x +5)
The attempt at a solution
Normally I would try to factor this into something like (x-1) (x+3) (That's an example completely unrelated to this problem.)
However, as no easy factors quickly occurred to me I did a run through of the...