Find the general solution of the first-order linear...

Transcript

00:01 Given this differential equation, we're asked to find the general solution.

00:04 So note that i'm going to start off with the auxiliary equation, and then basically we need to find the roots of this polynomial.

00:12 So how does it factor, in other words? and note that by the rational roots theorem, if it has a rational root, then it has to be a divisor of 220.

00:22 So the possible roots, rational roots are plus or minus one, plus or minus two, plus or minus four, dot, dot, dot, plus or minus 110, all the way up to plus or minus 220.

00:41 All right.

00:41 So if we try minus one or one, we notice that that's not going to be zero because this negative 220 is just too large of a number so that the whole thing would be zero.

00:56 However, perhaps two might do the job.

00:59 So we could test that and you can just plug it in or you can use polynomial division or synthetic division.

01:08 That's what i'm going to do.

01:11 So i dropped the one down to one times two, which is two and add these two.

01:17 So i get negative 18.

01:19 Multiply that by two.

01:20 Negative 36.

01:23 Add those two to get 83 and then multiply that two to get 166.

01:30 Add those two and get negative 54.

01:34 Ok, so this isn't zero.

01:36 Therefore, two is not a root of this polynomial.

01:41 But i can try four.

01:43 So just listing the coefficients.

01:47 So this is one one nine negative two two zero...

Question

Find the general solution of the first-order linear differential equation $y' - (\ln x)y = 6x^2$. y(x) = Note: Use $C$ for the arbitrary constant.

Question

Find the general solution of the first-order linear differential equation $y' - (\ln x)y = 6x^2$. y(x) = Note: Use $C$ for the arbitrary constant.

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