Calculus II PASS Week 11

Week 6

100

Find the solution of dx/dt = -7x, where x=3 when t=0

x(t) = 3e^-7t

200

Determine the general solution for: d²y/dx² - (6)dy/dx - 8y = 0

y(x) = Ae^8.83x + Be^-2.83x

300

Solve dy/dx + 6y = e^-3x, given that y(0) = 1

y(x) = (2/3)e^-6x + (1/3)e^-3x

400

Find the general solution y(t) of a system described by: dy/dt + 4y = 2 cos (t), where y(0) = 1

y(t) = (9/17)e^-4t +(8/17)cos(t) + (2/17)sin(t)

500

If d²y/dt² + 4 dy/dt + 3y = 0 with initial conditions of y(0) = 4 and y'(0) = 0, what is y(t)?

y(t) = 6e^-t - 2e^-3t

Week 7 & 8

100

If f(t) = e^t/4, what is the Laplace transform?

F(s) = 1/(s-(1/4))

200

If f(t) = e^-5t + 3t, what is the Laplace transform?

F(s) = (1/(s+5)) + 3/s²

300

If f(t) = 3 sin (4t) + t⁴, what is the Laplace transform?

F(s) = 12/(s²+4²) + 24/s⁵

400

If f(t) = cos(7t)*e^-4t + 9, what is the Laplace transform?

F(s) = ((s+4)/(s+4)² + 7²) + 9/s

500

If F(s) = (5s - 20)/(s-6)(s+2), solve for f(t)

f(t) = (5/4)e^6t + (15/4)e^-2t

Week 9

100

Find the partial derivative with respect to x of: (-cos(6x))/6

sin(6x)

200

Find the partial derivative with respect to t of: 3t sin 2x

3 sin 2x

300

Find u(x,y) if the partial derivative with respect to x is 3 cos (4y) + 2 and the partial derivative with respect to y is -12x sin (4y) + 15

u(x,y) = 3x cos 4y + 2x + 15y + C

400

Find all m such that y(x,t) = Acos(x)*e^mt is a solution of 3 ∂²y/∂x² = -∂y/∂t

3, therefore: y(x,t) = Acos(x)*e^3t satisfies the conditions.

500

For u(x,t) = cos(x)sin(at), find ∂²u/∂x² and ∂²u/∂t², and find values for 'a' if ∂²u/∂x² = (36)∂²u/∂t²

a = +- 1/6