Toronto Math Forum
MAT2442019F => MAT244Lectures & Home Assignments => Chapter 4 => Topic started by: jeyara85 on November 13, 2019, 10:42:22 PM

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

Finding the characteristic equation for higher order equations is very similar to the second order case.
If we have the differential equation $a_ny^{(n)} + a_{n1}y^{(n1)} .... + a_1y' + a_0y = 0$ then the characteristic equation is $a_nr^n + a_{n1}r^{n1} .... + a_1r + a_0 = 0$.
To find the roots, we can use the fact that the product of the roots must be $a_0$ to help guess the roots.

Yes, because for equations given they could be found easily

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

If any of the $a_i$'s are not constant, then we cannot use the method above. Nonconstant 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.