Differential Equations - Reduction of order

[Pinedas42]

General Question. How would I recognize when to do a full reduction of order versus using the formula y2=y1 integral(e^integral(P(x)dx)/y1^2)

So far I only know by when the homework problem set, specifies to use the formula or the reduction of order. However I want to know if I can tell somehow that the formula would not work for a specific problem.

I've been doing some practice problems and there are some that definitely cannot use the formula as I work through it so then I switch to reduction fo order. But there are some problems where I've done both ways out of curiosity and in both ways it resolves all the way, but I get two different solutions for y2.

Thank you!

 

[lippyka]

The best way to determine when to use the reduction of order method versus the formula is to look at the specific differential equation you are trying to solve. If the equation has terms that are already of higher order than two, then you may need to use the reduction of order method. Alternatively, if the equation only has terms up to second order, then the formula may work and you can try to employ it. The only way to determine which method will produce the correct solution is to try both methods and verify which one produces the correct result.