Wikipedia:Reference desk/Archives/Mathematics/2020 March 30

= 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 t1 − t0 = ..., 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
 * Separation of variables
 * -- ToE 18:53, 30 March 2020 (UTC)


 * Yeah, basically separate the variables so you have $${1\over bP^2}dP = dt$$, 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)