FP2 Chapter 4 – First Order Differential Equations Dr J Frost Last modified: 6 th August 2015

Intro The rate of temperature loss is proportional to the current temperature.

C4 recap Slap and integral symbol on the front!

Examples Why is it called the general solution? We have a ‘family’ of solutions as the constant of integration varies. ? So the ‘family of circles’ satisfies this differential equation. ?

Examples ??

Particular solutions Just like at the end of C1 integration, we can fix to one particular solution if we give some conditions. Some people (including myself) like to sub in as soon as possible before rearranging. ?

Exercise 4A

Using reverse product rule Quickfire Questions: ? ? ? ? ? ?

Using reverse product rule Test Your Understanding ? ? ?

But what if we can’t use the product rule backwards? Then multiplying through by the integrating factor: Then we can solve in the usual way: ? ? ?

Proof that Integrating Factor works ? ? ?

You could skip to here provided you don’t forget to multiply the RHS by the IF. What shall we do first so that we have an equation like before? STEP 2: Determine IF STEP 3: Multiply through by IF and use product rule backwards. STEP 4: Integrate and simplify. ? ? ? ?

Test Your Understanding FP2 June 2011 Q3 ?

Exercise 4C

Making a substitution Sometimes making a substitution will turn a more complex differential equation into one like we’ve seen before. ? ? ? ?

Another with an IF ?

Test Your Understanding FP2 June 2012 Q7 a ? b ? c ?

Exercise 4D/E