MAT244--2019F > Chapter 4



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.

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

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.


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