Question

Solve the differential equation 14y'= x + y
by making the change of variable u = x + y.

Answer 1

\[\large 14\color{royalblue}{\frac{dy}{dx}}=\color{orangered}{x+y}\]

\[\large \color{orangered}{u=x+y}\]\[\large \frac{du}{dx}=1+\frac{dy}{dx} \qquad \rightarrow \qquad \color{royalblue}{\frac{dy}{dx}=\frac{du}{dx}-1}\]

Answer 2

Understand how to plug in the substitution? :) I color-coded it to make it a little easier.

Answer 3

Sorry but I am still confused. I understand where everything came from but what do i do from where you left off? Do I write 14du/dx-1=u?

Answer 4

Yes very good! :)

We can use prime notation if it's a little easier to understand,\[\large 14(u'-1)=u\]

Answer 5

From here, distribute the 14 to each term in the brackets.
Then you can solve it either by separation of variables, or by getting it into standard form and finding an integrating factor.

Answer 6

I hope this step wasn't too confusing, \(\large \frac{du}{dx}=1+\frac{dy}{dx}\)

It's the same as \(\large u'=1+y'\)

I was just trying to emphasize that we're taking the derivative with respect to x. That's what the Leibniz notation was showing us.

Answer 7

No, I understand that. I just have a hard time solving them on my own. Thanks

Answer 8

Cool c: Lemme know if you get stuck.