How to solve this Diff Eq (with multiple terms) $$\frac{dy}{dx}=7xy$$
I know this turns into
$$\frac{dy}{y}=7xdx$$
.....etc.
But, how do you solve the following:
$$\frac{dy}{dx}=7x+y$$
Not sure how to seperate the parts to respective sides.

Here is another approach (guessing, otherwise known as undetermined coefficients).
We call this part the complementary solution, we have 
$$y' - y = 0$$
Do you  know a function that will provide a zero when differentiated and subtracted from itself?
Choose $y_c(x) = c e^x$.
We now have what is called the particular part of $y' - y = 7x$
Lets guess at $y_p = ax +b$ and substitute back in:
$$y' - y = a - (a x + b) = 7 x \implies a = b = -7$$
The final solution is the linear combination of complementary and particular, hence:
$$y(x) = y_c(x) + y_p(x) = ce^x -7x -7$$
As a check, substitute back into ODE and see you get equality.
We could have used various methods to solve this that include integrating factor, linear substitution, Laplace Transform, exact equation, series methods, Picard Iteration...
