Difference between separable and linear? Differentials

Question

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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?

score 2 · Answer 1 · answered Mar 12 '14 at 05:37

2

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.)

answered Mar 12 '14 at 05:37

Eric Towers

67,037

As a rule of thumb: $a(x)b(y)$ is always separable, and $a(x)+b(y)$ is often not. – Eric Stucky Mar 12 '14 at 05:39
@EricStucky I like that idea. So would C, D, F, G, H be separable? – user3042929 Mar 12 '14 at 05:41
@user3042929: Do you have a rule to split $\tan(xy)$ into something containing only $y$s and something containing only $x$s (and likely these two things are multiplied, not added.)? – Eric Towers Mar 12 '14 at 05:43
@EricTowers We do in the back of our textbook yes. But not typically allowed to use them on quizzes or exams. – user3042929 Mar 12 '14 at 05:45
@user3042929: Are you sure there's a way to (algebraically) split $\tan(xy)$ into ($x$ part)($y$ part) in the back of the textbook? – Eric Towers Mar 12 '14 at 05:52

score 0 · Answer 2 · answered Feb 14 '21 at 02:03

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.

Difference between separable and linear? Differentials

2 Answers2