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March 30

ODE

I have an equation of the form dP/dt = bP² where b is a constant.

How do I find the time taken for P to change from P0 to P1 (both are positive)?

Thank you

2A01:E34:EF5E:4640:35D3:546B:6E11:13A9 (talk) 17:07, 30 March 2020 (UTC)

Step 1:

Solve the differential equation. When you do this, you will first get an equation defining an algebraic relation between P and t. The equation should also have an constant of integration. You can solve this equation for P, giving you a formula for P as a function of t. However, for performing the next steps, this is not actually necessary.

Step 2:

Create two new equations from the one you have. In one, substitute P₀ for P and t₀ for t. In the other, do the same but now with P₁ fand t₁.

Step 3:

Manipulate these equations to get an equation of the form Template:Nowrap, in which there is no t in the right-hand side. The constant of integration should also disappear in the process, otherwise you did something wrong. Then the right-hand side is a formula for the time taken for the change.

If you get stuck, let us know.  --Lambiam 18:36, 30 March 2020 (UTC)

To be honest I was stuck as soon as I realised it was a differential equation. Numerical Recipes, my usual "go to" for this kind of thing wasn't much help. 2A01:E34:EF5E:4640:35D3:546B:6E11:13A9 (talk) 18:56, 30 March 2020 (UTC)

Relevant links for step 1:

Non-linear differential equations#Ordinary differential equations

Separation of variables

-- ToE 18:53, 30 March 2020 (UTC)

Yeah, basically separate the variables so you have $\frac{1}{b{P}^2}dP=dt$, then integrate both sides and solve for P. 2601:648:8202:96B0:E0CB:579B:1F5:84ED (talk) 20:02, 30 March 2020 (UTC)

-2(1/bP1 - 1/bP0) = t1 - t0. Is that it? 78.245.228.100 (talk) 20:28, 30 March 2020 (UTC)

Very close. Check your integration again. -- ToE 21:06, 30 March 2020 (UTC)

-1(1/bP1 - 1/bP0) = t1 - t0? 2A01:E34:EF5E:4640:35D3:546B:6E11:13A9 (talk) 21:14, 30 March 2020 (UTC)

Well done! -- ToE 21:23, 30 March 2020 (UTC)

Trivial errors easily slip in. Here is a tip. After integrating (or solving a differential equation in general), check the solution by taking derivatives and substituting all in the equations you started with. That way you'll catch most errors.  --Lambiam 06:10, 31 March 2020 (UTC)