Think of 4 useful integrals fromCore 2 and Core 3

Recap from Core 2,

Example 1

Find the general solution to the differential equation dy = x3

dx y

Given that y = 4 when x = 0, find the particular solution

Example 2

Find the general solution to the differential equationdy + x = xy

dx

Given y = 5 when x = 0, find the particular solution

Solving differential equations:

1). Separate the variables, y onto one side, x onto the other side

2). Integrate both sides

3). Substitute the limits in to find the constant

4). If possible, rearrange to make y the subject

Some for you:

1.

2.

3.

We have the differential equation,

dy = xy2 and the conditions y = 1 when x = 0dx

Show that the particular solution is y = 2

2 – x2

Why is this maths useful? This example is on Page 72 of your textbook:

A water tank is filled in such a way that the rate at which the depth of the water increases is proportional to the square root of the depth.

Initially the depth is 4m. After a time t hours the depth is h metres

a). Show √h = 0.5kt + 2

b). Given h = 16 metres after t = 6 hours, find the value of k

c). Find the time taken to fill the tank to a depth of 36m

A few of you struggled with this in the test before half term

Recap:

We saw last week how differential equations can be solved by separating variables and then integrating.

They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering.

They can describe exponential growth and decay,the population growth of species or the change in investment return over time

The letter k is often used as the constant

An bee population grows at a rate that is proportional to its population.ie. if there is a bigger population, it grows quicker

a). Using a differential equation, show that is population may be modelled by the equation P = Aekt , where t is time

b). If the initial population is 450, find the value of A

c). After 3 days, there are 1800, find the value of k to 3.s.f.

d). How many will there be after 6 days?e). When will the bee population reach 20,000?

An equation of the form y = aebt (where a and b > 0)represents exponential growth.

An equation of the form y = ae-bt (where a and b > 0)represents exponential decay.

A different population can be modelled by:

P = 40 + 10e-2t

a). What is the initial population?

b). Is the population growing or declining?

c). What happens as time goes on?

d). Sketch a graph showing the population

e). Will the population become extinct?

An equation of the form y = aebt (where a and b > 0)represents exponential growth.

An equation of the form y = ae-bt (where a and b > 0)represents exponential decay.

The equation y = c + ae-bt (a and b > 0) represents a process in which the value of y gets closer and closer to c as t → ∞

Recap from Tuesday

RecapWhere P is population and t is time,

P = 4000 + 5ekt

a). Given that after 3 years, there are 4430 rabbits, work out theconstant k to 3.s.f.

b). Give an estimate of the population when t = 5

c). What is the rate of growth when t = 1?

d). In real life, why would there be a limitto this population?

I put £500 into my bank account, there is an interest rate of 8% /year

a). Explain why M = 500(1.08)t , where M is amount of money and t is time.

b). How much money will I have after 3 years?

c). What is the rate of growth after 3 years?Why is this harder to calculate than the previous example?

Pi is the 16th letter of the Greek alphabet just as p is the 16th of our alphabet.

Pi is an irrational number (which means that is cannot be represented exactly by a fraction). This was first proved in the 18th century.

It was chosen for being the first letter of the Greek word for perimeter.

In 2005, Lu Chao of China set a world record by memorising the first 67,890 digits of pi.

Albert Einstein was born on Pi Day: March 14, 1879.

In the OJ Simpson trial in 1995, doubts were raised about the reliability of one witness when he got the value of pi wrong

Pi Day is celebrated on March 14th (3/14) around the world