Laplace Transforms and Analytic Geometry Notes

Assignment - Laplace Transforms and Analytic Geometry




1. 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 ).