Linear differential equations are those which can be reduced to the form Ly=f, where L is some linear operator. Your first case is indeed linear, since it can be written as: ((d^2)/(dx^2)-2)y=ln(x) While the second one is not. To see this first we regroup all y to one side: y(y'+1)=x-3 then we simply notice that the operator ymapsto g(y)=y(y'+1) is not linear (for example we can take two functions y_1 and y_2 and notice that g(y_1+y_2)ne g(y_1)+g(y_2)).

If the equation would have had ln(y) on the , that also would have made it non-linear, since natural logs are non-linear functions. Remember that this has its roots in linear algebra: y=mx+b. You can analyse functions term-by-term to determine if they are linear, if that helps. The first time a term is non-linear, then the entire equation is non-linear.
Remember that the xs can pretty much do or appear however they want, since they're independent. Which means if you can't tell just by glancing, try to group all your y terms to one side and then analyse them. Makes it much easier.
See, I was also overthinking this, but realised you have to go back to those definitions we're given.
Two criteria for linearity:
1.The dependent variable y and its derivatives are of first degree; the power of each y is 1. (dy)/(dx);
2. Each coefficient depends only on the independent variable x.
yy' makes it nonlinear as has been said, because that coefficient on y' is not x. Had that coefficient been a constant, you would have been correct to call it linear, since constants can be functions of x. Like, f(3)=x. Its graph is a line, i.e. linear function.