Question

solve the differential equation
y'=x+5y

Answer 1

this is linear right?

Answer 2

for whatever reason I wanna take the derivative but that's not what I'm supposed to do . Yes linear

Answer 3

\[\LARGE \frac{dy}{dx} - 5y = x\]
use integrating factor
\[e^{\int -5dx} = e^{-5x}\]
then do that \[y(I.F.) = \int Q(I.F.)dx\] thingy
\[\large y(e^{-5x}) = \int xe^{-5x}dx\]
where did you get confused?

Answer 4

ok I have never done this before

Answer 5

e?

Answer 6

ok lets see what's going on... we have \[e^{\int -5dx}\] and \[e^{ -5x}\]

Answer 7

oh you havent? it's just \[\frac{dy}{dx} + Py = Q\] where P and Q are functions of x then you solve for integrating factor
\[e^{\int Pdx}\]
once you have solved the integrating factor, you use the equation
\[y(I.F.) = \int Q(I.F.)dx\]
where I.F. means integrating factor

Answer 8

are you not familiar with integrating factors in linear DE?

Answer 9

I'm not sure, P in this case is 5x

Answer 10

no it's just 5

Answer 11

ok

Answer 12

wait it's -5

Answer 13

how about Q 1?

Answer 14

Q would be x
\[\large \frac{dy}{dx} + Py = Q \implies \frac{dy}{dx} - 5y = x\]

Answer 15

ok