4.2

[jeyara85]

I was wondering if we will be expected to find the characteristic equations of higher-order equations (3rd,4th, ..., nth), or will we be provided the characteristic equation in the exam.

[david]

Finding the characteristic equation for higher order equations is very similar to the second order case.

If we have the differential equation $a_ny^{(n)} + a_{n-1}y^{(n-1)} ....  + a_1y' + a_0y = 0$ then the characteristic equation is $a_nr^n + a_{n-1}r^{n-1} .... + a_1r + a_0 = 0$.

To find the roots, we can use the fact that the product of the roots must be  $a_0$ to help guess the roots.

[Victor Ivrii]

Yes, because for equations given they could be found easily

[ZeCheng Wu]

what if a_(n) is not a constant, like x^2 for example

[david]

If any of the $a_i$'s are not constant, then we cannot use the method above. Non-constant coefficient differential equations are generally harder to solve. We discussed a few methods in class such as reduction of order or using the Wronskian, but both methods require already knowing one solution.