someone plssss help!!!! (1 Viewer)

[johnncenaa]

hey dudes, i'm having major struggles with exponential growth and decay.

I don't mind so much the working out of the problems, but I just don't understand why they ask questions such as eg.
Its temperature, T°C, after t hours in the oven satisfies the equation dT/dt= −k (T − 190)
Show that T = 190 – 185e^ –kt satisfies both this equation and the initial condition.

what is the purpose of it?
Would someone mind explaining the theory behind it bc the textbook is really general+ youtube is giving me nothing!

Thxxx so much in advance!!

 

[Paradoxica]

hey dudes, i'm having major struggles with exponential growth and decay.

I don't mind so much the working out of the problems, but I just don't understand why they ask questions such as eg.
Its temperature, T°C, after t hours in the oven satisfies the equation dT/dt= −k (T − 190)
Show that T = 190 – 185e^ –kt satisfies both this equation and the initial condition.

what is the purpose of it?
Would someone mind explaining the theory behind it bc the textbook is really general+ youtube is giving me nothing!

Thxxx so much in advance!!

Make the general substitution U = T-190

Observe dU/dt = dT/dt, since they differ by a constant.

Then you are solving the homogenised differential equation

dU/dt=-kU

dU/U = -kdt

log(U) = C-kt

Back-substitute U to obtain

log(T-190) = C-kt

T-190 = Ae^-kt, where A = e^C

Use initial conditions to obtain the value of A, and that is your non-trivial solution.

In 2/3U, the non-trivial solution to the differential equation is usually given to you, in 4U they expect you to find it yourself.

 

[leehuan]

But as for the purpose? In the real world, you'd have to do what Paradoxica did.

You don't just KNOW that the equation T=190-185e^(kt) is supposed to be a solution. How, in the real world, can Newton's law of cooling be analysed conveniently without actually going through the process of solving the differential equation and getting that expression out in the first place?


Things like Newton's Law of Cooling are defined using the differential equation form. NOT, the C + Be^(kt) form

 

[johnncenaa]

Thankyou leehuan, you are a talent where explaining is concerned!!!

Just with your second comment, are you saying the reason we get asked all the time to show that C+Be^kt is a solution is just to reiterate that we start with the differential? Also is C+Be^kt called a solution because of the nature of the exponential function, where its differential is the same as the original function?

 

[InteGrand]

Thankyou leehuan, you are a talent where explaining is concerned!!!

Just with your second comment, are you saying the reason we get asked all the time to show that C+Be^kt is a solution is just to reiterate that we start with the differential? Also is C+Be^kt called a solution because of the nature of the exponential function, where its differential is the same as the original function?

It's a solution because it satisfies that equation (called a differential equation). And the reason they ask you to show that's a solution is also to basically tell you what the solution is (since it is usually needed for later parts); they can't ask "solve the differential equation", because solving that differential equation from scratch isn't required by the HSC syllabus, so instead they give you the solution and make you verify it is indeed a solution.

What Paradoxica did is actually solve the differential equation from scratch, but you don't need to do that in the HSC (not are you expected to); you just need to sub. the function they give you into the differential equation and show the equation is satisfied (as well as the initial condition in this particular question).