Solving Variable Coefficient ODEs: Step-by-Step Guide for Your ODE Exam

[cefarix]

ODE Exam! A simple ODE...please help!

I've got a midterm exam later this morning on ODEs. I know how to solve second-order and higher ODEs with constant coefficients, but what about variable coefficients??

Can someone please walk me step-by-step through the solving of:
y'' + A(x)y' + B(x)y = f(x), where ' indicates a derivative with respect to x. I really need to know this before the exam (I'll study for it myself too of course, but it would be really helpful if someone also explained it to me.). Thank you very much in advance for taking the trouble to help!

 

[amcavoy]

I've got a midterm exam later this morning on ODEs. I know how to solve second-order and higher ODEs with constant coefficients, but what about variable coefficients??
Can someone please walk me step-by-step through the solving of:
y'' + A(x)y' + B(x)y = f(x), where ' indicates a derivative with respect to x. I really need to know this before the exam (I'll study for it myself too of course, but it would be really helpful if someone also explained it to me.). Thank you very much in advance for taking the trouble to help!

Well, it looks like you've already had your exam. Have you heard of Reduction of Order?

Alex

 

[HallsofIvy]

In general there is NO simple general way of solving linear equations with variable coefficients (apmcavoy's suggestion of "reduction of order" works to reduce a 2nd order equation to a first order IF you already know a solution). The most general way of solving such an equation is to use a series solution. That depends strongly on the specific functions involved- and is much too complicated to be given here. If you are supposed to take a test over such problems then surely it is in your textbook and/or has been gone over in class?!