Find the general solution yt which solves the problem below...

Find the general solution, $y(t)$, which solves the problem below, by the method of integrating factors. $4t \frac{dy}{dt} + y = t^4$ Find the integrating factor, $u(t) = t^{0.25}$ and then find $y(t) = \frac{5t^3}{4t} + \frac{C}{4t}$ (use C as the unknown constant.)

Transcript

00:01 Okay, in this question we are given a differential equation.

00:04 5t times by dt plus y being t to the power 4.

00:15 Okay, to find the integrating factor we want to first write this as dy dt plus pt y equal to qt.

00:28 And if we want to write it, this is the standard form.

00:32 So the first thing for us to do is to kill the 5t before dy dt.

00:41 That means we can write this as dy dt is equal to t, oh sorry, sorry, plus 1 over 5t, 5 being t to the power of 3 over 5.

01:02 Ok, once we can write our differential equation, i mean the first order differential equation, if it stands on 4, then we know the integrating factor must be equal to e to the power of dt, where t equals pt dt.

01:20 So let's first deal with this guy.

01:23 We have nt to the t of 1 over 5t dt, which is equal to 1 over 5 times lorin, the absolute value of t.

01:38 I mean the integrating factor plus some constant c.

01:43 I mean the nt to the t for this term is equal to something with respect to constant c, but as this integrating factor works for any constant c, that means we just need to find the most simple one.

02:02 We can just choose ut to be equal to 1 over 5 log nt e to the power, okay? it's better for us to express ut as this guy.

02:29 I mean we only need to find the simplest integrating integrating factor.

02:35 Then we find 1tu, the integrating factor is equal to t to the power 1 over 5.

02:43 Okay, then multiply this guys...

Question

Find the general solution, $y(t)$, which solves the problem below, by the method of integrating factors. $4t \frac{dy}{dt} + y = t^4$ Find the integrating factor, $u(t) = t^{0.25}$ and then find $y(t) = \frac{5t^3}{4t} + \frac{C}{4t}$ (use C as the unknown constant.)

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