Taking a Differential Equations Class; Can't remember/understand anything.

**MrAsianMan** · August 24th 2010, 09:29 PM

Hi,
I'm currently enrolled in a Differential Equations class at my local community college. I'm trying to get through the first problem, but I don't even remember/know what I'm doing.

Right now, I'm having trouble with classification by linearity. I can't tell (without guessing and looking at the back of the book

) whether a DE is linear or non-linear.

An problem I'm having trouble understanding goes as:

Determine whether the given first-order DE is linear in the indicated dependent variable by matching it with the first differential equation given in (7)
$[(y^2)-1]dx + x dy = 0; in y; in x$ .

The Answer Book said:
Writing the DE in the form x(dy/dx) = Y^2 = 1, we see that it is nonlinear in y because of y^2. However, writing it in the form of (y^2 - 1)(dx/dy) + x = 0, we see that it is linear in x.

I can't remember anything really about dx or dy and how the question got to the solution. If anyone can direct me to the subject online I should review (I have the 6e stewart book if anyone else knows/uses it).

I'm sorry for my ignorance; hope someone can help.

**Haven** · August 25th 2010, 10:01 PM

Linear basically means it has the same properties as a line. The easiest classification of this is that the equation only contain terms that are singular powers (i.e., $x^1$ ).

So $x^2+1$ is not linear
$\sqrt{x} + x$ is also not linear.
however, $3x+2y$ is linear. (no powers, exponentials, logarithms, roots, just plain old ordinary variables)

In general, if we haves variables $x_1 \dots x_n$ , then a linear equation is of the form.

$c_1x_1 + \dots + c_nx_n = d$

Where $c_1 \dots c_n$ and d are all constants.

Now a first order ordinary differential equation is always of this form

$\frac{dy}{dx} = f(y)$

where f(y) is some expression involving y, as a function of x.

So, $\frac{dy}{dx} = 3y$ is a differential equation.
and $\frac{dy}{dx} = e^y$ is also a differential equation.

We call a differential equation linear, if f(y) is linear. So the first one is linear, but the second one isn't.

How the book got it's answers was through simple algebra. In order to determine if the differential equation is linear in x, we needed the derivative of x on one side and an expression involving x on the other.

So, we do this $(y^2-1)dx + xdy = 0 \Rightarrow (y^2-1)dx = -xdy \Rightarrow (y^2-1)\frac{dx}{dy}= -x$

Since the right hand side is a linear expression, we say the differential equation is linear in x.

However, similar algebra tells us that the differential equation is NOT linear in y.
i.e., $(y^2-1)dx + xdy = 0 \Rightarrow (y^2-1)dx = -xdy \Rightarrow x\frac{dy}{dx}= 1-y^2$ . Since the right hand side is not linear, the differential equation is not linear.

Thread: Taking a Differential Equations Class; Can't remember/understand anything.

LinkBack

Thread Tools

Display

Taking a Differential Equations Class; Can't remember/understand anything.

Similar Math Help Forum Discussions

Help me understand and remember this Theorem

taking trig class and forgot how to do this problem.

Differential Equations-Application of 1st order differential equations 1

Don't understand this differential equation

taking intro to ordinary differential equations; need some help

Search Tags