Solving Homogeneous First Order Differential Equations - how to stay on track!
Tom’s latest video is a satisfying step by step guide to solving a “homogeneous first order differential equation”. Don’t be put off by this long & intimidating name however, like all of Toms videos it goes at a pace that is easy to follow for any maths student (or Tom Rocks Maths super fan who has kept a keen eye on previous episodes in the Oxford Calculus series) whilst also giving the brain a bit of a work out!
As any experienced equation solver will know, while the enigma of a first order differential equation is solvable, there are many classic errors that could trip you up along the way and make those numbers, letters and symbols merge into an ODE nightmare. Here are 5 handy tips to keep you on track as you navigate Tom's latest challenge
1. The “Minus Sign”
That dreaded ‘-’. So small and easy to overlook but with the power to ruin your equation, often without you even noticing until you're down a rabithole of substitutions and differentials looking for a way to simplify your overloaded fraction. Just take that second to double check and be sure that each ‘+’ and ‘-’ is definitely correct, it’s going to save you more time overall.
2. Functions vs constants
As you differentiate, integrate and rearrange your function it’s natural to try and look for simplicity and shortcuts. It sure is quicker and neater to write a function like y(x) as ‘y’, but if ever you differentiate by x, treating this y as a constant rather than a function will really screw up your work! To avoid this catastrophe be like Tom and keep your v’s, x’s & y’s for functions and a’s, b’s & c’s for constants.
If you find your fraction getting longer, your integrals getting more complex and just generally running out of space on your page it may be time to make your workings simpler using a substitution. Take care with any differentiating but otherwise substitutions are the perfect gift for your equations.
4. Know your integrals
Every calculus student (or general equation fanatic) knows they need a good arsenal of standard integral solutions to get them through an ODE. It may be worth a quick differentiation and integration refresher (Tom has plenty more content that can help you with this) so you don’t get stumped by integrating a ‘1/x’ once it gets more complicated. And remember - don’t forget that +c.
5. Check your answer!
As is often the case in calculus there are many steps to find that end solution and when you finally get there it’s normal to want to go put your feet up and have a cup of tea from your favourite Tom Rocks Maths mug (which we don’t discourage). Once you’re refreshed and ready to go again, make sure to check your solution, in this case by differentiating with respect to x to see if you can get back to that original homogeneous first order differential equation!