Solving a first order differential equation

In summary, the conversation discusses solving a differential equation using numerical methods and the Lambert W function. It also mentions rearranging an equation with P on one side and everything else on the other side.
  • #1
zak8000
74
0
hi

the differential equation i am attempting to solve is:

[tex] \frac {dP} {dx} = \frac {gP} {1+P/Psat} [/tex]

here is what I have done:

[tex] \frac {dP} {dx} = \frac {gP*Psat} {Psat+P} [/tex]

divide both sides by [tex] \frac {Psat+P} {gP*Psat} [/tex]

to get:
[tex] \frac {Psat+P} {P*Psat} \frac {dP} {dx} =g [/tex]

[tex]\int \frac {Psat+P} {P*Psat} dp = \int gdx [/tex]

[tex]\int \frac {dp} {P}+ \int \frac {dp} {Psat} =gx+c [/tex]
[tex]ln(P)+ \frac {P} {Psat} =gx+c [/tex]

now how do i rearrange P on one side with everything else on the other side
 
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  • #2
By realizing, P = ln(eP)
 
  • #3
Hi zak8000 !

In practice, the equation ln(P)+P/Psat = gx+e is solved thanks to numerical methods.
The analytic solution requieres a special function W(X), namely the Lambert W function.
P/Psat = W(X) with X=exp(gx+e)/Psat
 

1. What is a first order differential equation?

A first order differential equation is an equation that relates a function to its derivative. It can be written in the form of dy/dx = f(x,y), where y is the dependent variable and x is the independent variable. Solving this equation means finding the function that satisfies it.

2. What is the process for solving a first order differential equation?

The process for solving a first order differential equation involves finding an integrating factor, multiplying both sides of the equation by it, and then integrating both sides. This will yield the solution to the equation in the form of y = f(x).

3. What is an initial value problem?

An initial value problem is a type of first order differential equation that includes an initial condition. This means that the equation has a given value for the dependent variable at a specific value of the independent variable. The solution to an initial value problem will be a specific function that satisfies the equation and the given initial condition.

4. Can all first order differential equations be solved analytically?

No, not all first order differential equations can be solved analytically. Some equations may require numerical methods or approximation techniques to find a solution. However, many commonly encountered first order differential equations can be solved analytically using the methods mentioned above.

5. How do I know if my solution to a first order differential equation is correct?

You can check the validity of your solution by substituting it back into the original equation. If it satisfies the equation, then it is a valid solution. Additionally, you can also compare your solution to known solutions or use a computer program to verify the accuracy of your solution.

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