Differential equations

i'm stuck on this question so can anyone help me please? :smile:

An alternative model gives the differential equation dx/dt = 1.4t - 0.5x for the same values of t (these values were given in the first part - it works for values of t up to 10). Verify that x = 13.6e^(-0.5t) + 2.8t - 5.6 satisfies this differential equation. Verify also that when t = 0 this function takes the value 8.

i managed to get to a point (by getting the x's and t's on their own and then integrating each side) where i have: 2lnx + k = 0.7t^2 but i'm not sure if its right or where to go from there

Reply 1

Gaz031

15

Juncture

i'm stuck on this question so can anyone help me please? :smile:

An alternative model gives the differential equation dx/dt = 1.4t - 0.5x for the same values of t (these values were given in the first part - it works for values of t up to 10). Verify that x = 13.6e^(-0.5t) + 2.8t - 5.6 satisfies this differential equation. Verify also that when t = 0 this function takes the value 8.

i managed to get to a point (by getting the x's and t's on their own and then integrating each side) where i have: 2lnx + k = 0.7t^2 but i'm not sure if its right or where to go from there

It doesn't look quite right but notice you don't actually have to solve the differential equation from scratch - you just need to verify the given solution. i.e. calculate dx/dt and 1.4t-0.5x and check they agree.

Reply 2

nota bene

15

The verification is just to plug in 0 into the answer...

We'd need the initial conditions to get a particular solution for the DE though...

edit: As Gaz says, you are not actually asked to solve the DE...

Reply 3

Juncture

OP

16

nota bene

The verification is just to plug in 0 into the answer...

I knew someone was going to say that, that was not the bit i was stuck on i just wrote it on the end for completeness

i still don't understand how the x = (what it says above) bit can be related to the dx/dt = 1.4t - 0.5x bit

Edit: I think I get it now :biggrin:

Reply 4

SunGod87

7

x = 13.6e^(-0.5t) + 2.8t - 5.6
dx/dt = (-1/2)(13.6e^(-0.5t)) + 2.8
dx/dt = -6.8e^(-0.5t) + 2.8

1.4t - 0.5x = 1.4t - 0.5(13.6e^(-0.5t) + 2.8t - 5.6)
1.4t - 0.5x = 1.4t - 6.8e^(-0.5t) - 1.4t + 2.8
1.4t - 0.5x = -6.8e^(-0.5t) + 2.8

Hence dx/dt = 1.4t - 0.5x

x(t) = 13.6e^(-0.5t) + 2.8t - 5.6
x(0) = 13.6e^0 + 2.8*0 - 5.6
x(0) = 13.6 - 5.6 = 8

Differential equations

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