Explaining Linear Equations: 2ty' + 4y = 2t^3

[suspenc3]

Hi, I am finding some of this confusing, can someone explain this?

so I undersand that $$xy' + y = (xy)'$$

lets say that I have $$2ty' + 4y = 2t^3$$, what is (xy)'?

would it just become $$d/dx(2ty) = 2t^3$$?

 

[StatusX]

You're mixing up x and t. If you just want to solve:

$$2t \frac{dy}{dt}+4y=2t^3$$

then you should use integrating factors. That is, find a pair of functions f(t) and g(t) such that (substituting back y' for dy/dt):

$$f(t) (2t y'+4y)= \frac{d}{dt} (g(t) y)$$

Right away you can see that 2tf(t)=g(t), and then you can get a simple ODE to solve for g(t). Now you multiply across in the original equation:

$$f(t) (2t y' +4y)=\frac{d}{dt} (g(t) y)=f(t) 2t^3$$

and then you just need to integrate. Note that the case (xy)' you describe first is another example of integrating factors, in that case with g(x)=x. In this case, g will be different.