Class notes
Laplace Transforms and Analytic Geometry Notes
- Course
- Institution
Detailed worked examples demonstrating how to solve Laplace Transforms and Analytic Geometry.
[Show more]
Seller
Follow
ahmedbraimah
Content preview
Assignment - Laplace Transforms and Analytic Geometry1. Use the table of transforms to find the inverse Laplace transform of
s
.
s2 − 2s + 17
[7 marks]
2. Find the angle between the plane P ,
−x + 3y + z = 5 ,
and the line L,
x = 3t − 1 , y = t + 2, z = −t .
Also find the point where L crosses P . [7 marks]
3. For the curve given parametrically by r(t) = (t3 , cos πt, sin πt), determine an equation
for the tangent line to the curve at the point r(1) and, thinking of t as time, a linear
approximation to the acceleration a = d2 r/dt2 centred on t = 1. [7 marks]
4. Two parts of a forced heavily damped mechanical system move according to the system
of differential equations
dx1 dx2
= −k1 x1 − k3 (x1 − x2 ) + f (t) , = −k2 x2 − k3 (x2 − x1 ) .
dt dt
Initially the displacements x1 and x2 are zero.
Take k1 = k2 = 3 and k3 = 2 with f (t) = e−5t and set up the differential equations and
initial conditions. Use Laplace transforms to find x1 (t) and x2 (t). [9 marks]
Answers
1
1. et (cos 4t + sin 4t); 2. 0.091 radians (5.22◦ ), (5, 4, −2);
4
3. r = (1, −1, 0) + s(3, 0, −π), a ≈ (6t, π 2 , π 3 (t − 1));
4. x1 = 41 (e−3t − e−7t ), x2 = 14 (e−3t − 2e−5t + e−7t ).
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller ahmedbraimah. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
78998 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now