Core 4 - Differential Equations

Question 1

If the rate of a snowball melting. Is constant, form a differential equation in terms of surface area A

Answer 1

dA/dt = - k

Question 2

How can the chain rule be applied to differential equations?

Answer 2

If we need to find one parameter in terms of another, we can use chain rule,

e.g. if =4pi r^2, and given ds/dt=640, find dr/dt

dr/dt=dr/ds * ds/dt

Question 3

When solving a differential equation, which term must you NOT FORGET?

Answer 3

+C on the end

it is rarely zero, and is determined by initial conditions.

Question 4

When do you add +c to the integral?

Answer 4

As soon as possible, the find quickly to avoid confusion due to rearrangement

Question 5

How much rearranging do you need to do when solving a differential equation

Answer 5

As little as possible

Question 6

To find the dN/dt from an expression for N (originally in terms of t) in terms of N, what is the most efficient method?

Answer 6

Differentiate by chain/quotient/product rule to find differential in terms of t, then rearrange the original to get t in terms of N, then substitute and simplify.

Question 7

If the rate of change of the area of a stain =640 cm^2s^-1, what can you say about the rate of change of radius?

Answer 7

dA/dt=dA/dr*dr/dt

A=(pi)r^2 => dA/dr=2(pi)r
640=2(pi)r=dr/dt

Question 8

If y=Ae^-cos(t), given y=50 when t =pi, find A.

Note, this is unusual because t=/= 0.

Answer 8

Ae=50, so A=50/e.

Y=50e^-1*e^-cos(t)
Y=50e^-(1+cos(t))

Question 9

How can we use area formula to find a differential equation for rate of change of area of a disk if proportional to perimeter?

Answer 9

Rearrange area equation for r=, then sub into perimeter equation, to get in terms of A, then sub into dA/dt.