Differential Equation: Separation of Variables

In summary: 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.
  • #1
brikayyy
15
0

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 = 100ep/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!
 
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  • #2
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:
 
  • #3
ymlc said:
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!
 
  • #4
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?
 
  • #5
brikayyy said:
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.
 
  • #6
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.
 

1. What is the concept of separation of variables in differential equations?

The concept of separation of variables in differential equations involves splitting a multi-variable differential equation into several single-variable equations, which can then be solved independently. This technique is used to solve differential equations that have one dependent variable and multiple independent variables.

2. How do you solve a differential equation using separation of variables?

To solve a differential equation using separation of variables, you need to follow these steps:1. Rearrange the equation so that all terms containing the dependent variable are on one side and all terms containing the independent variables are on the other side.2. Integrate both sides of the equation with respect to the dependent variable.3. Integrate both sides of the equation with respect to the independent variable.4. Add a constant of integration to the resulting equation.5. Solve for the dependent variable.

3. What types of differential equations can be solved using separation of variables?

Separation of variables can be used to solve first-order and second-order differential equations. It is most commonly used to solve linear differential equations, but it can also be applied to nonlinear equations under certain conditions.

4. Can separation of variables be applied to partial differential equations?

Yes, separation of variables can be applied to some types of partial differential equations. However, it is not a universal technique for solving partial differential equations and may not work for all equations.

5. Is separation of variables the only method for solving differential equations?

No, separation of variables is not the only method for solving differential equations. There are various other techniques such as substitution, integrating factors, and power series that can be used to solve different types of differential equations.

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