Starter Solve the differential equation we came up with last lesson to model the following scenario.

Differential Equations 2 Particular Soltions Core 4 Maths with Liz

AIMS By the end of the lesson, you should be able to: solve first order differential equations by separating the variables find the particular solution to a first order differential equation by using given conditions

Example 1 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 time t hours, the depth is h m. (a) Write down a differential equation for h.

Example 1 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 time t hours, the depth is h m. (b) Show that , where k is a constant. Remember, k is simply a constant. We are still integrating with respect to t on the R.H.S. This is nearly what we want, but we still need to get rid of the constant, C. To do so, we must use more information from the given problem…

Example 1 This means that when time t = 0, the depth h = 4m. 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 time t hours, the depth is h m. (b) Show that , where k is a constant. This means that when time t = 0, the depth h = 4m. Substitute back into our solution:

Example 1 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 time t hours, the depth is h m. (c) Given that h = 16 when t = 6, find the value of k.

It takes 12 hours to fill the tank to a depth of 36m. Example 1 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 time t hours, the depth is h m. (d) Find the time taken to fill the tank to a depth of 36m. This is asking us to find t when h = 36. Keep in mind, we just determined that our constant k = 2/3. It takes 12 hours to fill the tank to a depth of 36m.

Exam Question June 2011, Q7

Exam Question June 2011, Question 7 To find k and C, we need to use the given information above.

Exam Question June 2011, Question 7 This means: When time t = 0, radius r = 60, and when time t = 9, radius r = 30. Letting t = 0 helped a bit, but we still have A remaining in the problem. Since we know A = 4𝝅 𝒓 2 , rewrite our equation as:

Exam Question June 2011, Question 7 This means: When time t = 0, radius r = 60, and when time t = 9, radius r = 30. Now, we just need to find k. Let t = 9, r = 30, and C = 14400𝝅. Substitute our new found k and C back into our equation we came up with way back when!

Exam Question June 2012, Q8

How do we integrate this? Exam Question June 2012, Q8 How do we integrate this? You must use integration by parts or substitution. Since we did by parts earlier in the lesson, let’s use substitution this time!

Exam Question June 2012, Q8 Integration by substitution: Let Since the 15x still remains, we must solve for x in our equation above and substitute: Substitute: Integrate both sides: Rewrite the equation in terms of x.

Exam Question June 2012, Q8 Almost done! We just need to use information given in the original question to find C. The question asked us to find t as a function of x, so we need to make t the subject (thank goodness!!!). This means that when time t = 0, the depth x = 1. Work this out in your calculator to save time!

Homework Core 4 Mock Exam Set homework over Differential Equations & Exponential Functions due Friday, 1st May. Test over topic will be held on Friday, 1st May as well. Upcoming topic: Functions defined Implicitly (Pg. 84 in Core 4 textbook) Core 4 Mock Exam will be held Friday, 4th May.