Differential Equation

Bit stuck on how to get the answer for exercise 3. Attached photo with my working. I’ve kind of got what I’m supposed to but don’t know how to get rid of the cos function so that I’m just left with the sine function.

And for exercise 4, would I just differentiate, set to 0 and then go from there?

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Reply 1

vc94

In ex.3, in your solution for y, aren't you missing some arbitrary constants?
In ex.4, isn't terminal velocity achieved when t tends to infinity?

Reply 2

username6352379

Original post by vc94

In ex.3, in your solution for y, aren't you missing some arbitrary constants?
In ex.4, isn't terminal velocity achieved when t tends to infinity?

Would it just be A and B in front of cos and sin respectively? Idk how to go from there to the solution they want.

I thought it would be when acceleration = 0 but I’m probably wrong.

Reply 3

mqb2766

Original post by kwikmaffs

Would it just be A and B in front of cos and sin respectively? Idk how to go from there to the solution they want.

I thought it would be when acceleration = 0 but I’m probably wrong.

Yes for A and B. The question asks for a/one solution, not the general solution so what (obvious) values for A and B would give what they want?

Terminal velocity would have dv/dt = 0, though as its name suggests that it would be the "terminal" value so as t -> inf? If you sketch the velocity curve / do some very simple reasoning about the fraction it should be clear (rather than doing calculus).

Reply 4

username6352379

Original post by mqb2766

So 1/2 and then minus one from the other? Thanks! Haven’t actually done second order differential equations yet but it appeared on the homework so it threw me a bit.

Reply 5

mqb2766

Original post by kwikmaffs

So 1/2 and then minus one from the other? Thanks! Haven’t actually done second order differential equations yet but it appeared on the homework so it threw me a bit.

If its
y ~ Asin + Bcos
then its just A=1, B=0?

Reply 6

username6352379

Original post by mqb2766

If its
y ~ Asin + Bcos
then its just A=1, B=0?

Oh yeah that works, sorry I watched a YouTube video where they added and minused 1 equations and multiplied by 1/2. Thanks :smile:

Reply 7

mqb2766

Original post by kwikmaffs

Oh yeah that works, sorry I watched a YouTube video where they added and minused 1 equations and multiplied by 1/2. Thanks :smile:

If the question had said "verify", it would simply have wanted you to substitute it into the ODE and verify the LHS was zero. If youre a bit unsure/rusty/... Its not a bad short exercise (rather than watching youtube) to verify that both
y = e^(-2mt)sin(4mt)
y = e^(-2mt)cos(4mt)
are solutions to the ODE, then deduce why
y = Ae^(-2mt)sin(4mt)y + Be^(-2mt)cos(4mt)
is also a (the general) solution to the linear ODE.