Differential Equation: Separation of Variables

[brikayyy]

Homework Statement

dL/dp = L/2, L(0) = 100. Find the solution to the differential equation, subject to the given initial condition. My textbook says the answer is L = 100e^p/2, but I don't know how to get that answer (or e for that matter).

Homework Equations

?

The Attempt at a Solution

dL/dp = L/2
L dL/dp = 1/2
∫LdL = ∫(1/2)dp
L²/2 = x/2 + C
L² = x + 2C
(0)² = 100 + 2C

Thanks for any help ahead of time!

 

[ymlc]

I found two mistakes in your procedure. In the second step of your solution, the L on the left side should divide dL.

Then, the way to apply the initial condition is to make x=0, L=100, and solve for C. You did x=100 and L=0. D:

 

[brikayyy]

I found two mistakes in your procedure. In the second step of your solution, the L on the left side should divide dL.

Then, the way to apply the initial condition is to make x=0, L=100, and solve for C. You did x=100 and L=0. D:

Oh, man. D: Thanks for pointing that out!

 

[brikayyy]

Now that I think about it, it looks like I'm supposed to use dy/dx = ky instead of what I did in my attempt?

 

[Dick]

Now that I think about it, it looks like I'm supposed to use dy/dx = ky instead of what I did in my attempt?

You don't need a formula, you can derive the result by separation of variables. But as ymlc pointed out you are making a mistake. You should have dL/L=(1/2)dp after separating. NOT LdL=(1/2)dp. Work from there.

 

[LCKurtz]

Now that I think about it, it looks like I'm supposed to use dy/dx = ky instead of what I did in my attempt?

That equation can be solved either by separation of variables or using linear equation theory and multiplying by an integrating factor. Or, for that matter, as a constant coefficient DE if you have had that method.