Difference between separable and linear? Differentials

Question

My understanding was that a separable equation was one in which the x values and y values of the right side equation could be split up algebraically. I tried this once before and got the wrong answer. Can someone help me?

Answer 1

Linear: No products or powers of things containing $y$. For instance $y'^2$ is right out.

Separable: The equation can be put in the form $\mathrm{d}y \left( \right.$expression containing $y$s, but no $x$s, in some combination you can integrate$\left. \right) = \mathrm{d}x \left( \right.$expression containing $x$s, but no $y$s, in some combination you can integrate$\left. \right)$.

(Technically, you don't have to be able to integrate the expression containing only $x$s or the expression containing only $y$s. However, if you can't, then you're not going to solve the equation.)

Answer 2

Consider the equation $\frac{dy}{dt} = ry$. This equation is a separable differential equations since we can rewrite this in the form of $\frac{dy}{y} = rdt$.

Consider the fact that this is also a linear equation since $\frac{dy}{dt} - ry = 0$ all the derivatives are attached to purely functions of t, and 0 is also a function of t. Also, y is raised to the power of 1.

On the other hand, take a look at the first equation in your picture. That equation is nonlinear since y is raised to the power of 2, but it is separable.