is there lenear euqation that..

transgalactic · #1 $Old$ November 1st, 2009, 07:08 AM

doesnt have constant coefficients?

Prove It · #2 $Old$ November 1st, 2009, 07:28 AM

Quote:

Originally Posted by transgalactic $View Post$

doesnt have constant coefficients?

Yes.

As long as you have (for lack of a better term) a "polynomial DE" of the form

$f_n(x) \frac{d^ny}{dx^n} + f_{n - 1}(x) \frac{d^{n - 1}y}{dx^{n - 1}} + f_{n - 2}(x) \frac{d^{n - 2}y}{dx^{n - 2}} + \dots + f_2(x)\frac{d^2y}{dx^2} + f_1(x) \frac{dy}{dx} + f_0(x) y = f(x)$ .

In other words, as long as the coefficients are nothing more than functions of x, the equation is still linear.

transgalactic · #3 $Old$ November 1st, 2009, 09:38 AM

can you give an example of some thing which is not a function of x?

Prove It · #4 $Old$ November 1st, 2009, 06:22 PM

E.g.

$x^2\frac{d^2y}{dx^2} + 3x^4\frac{dy}{dx} + 7xy = 8\sin{x}$ .

Do you see that the coefficients are nothing more than functions of x?

transgalactic · #5 $Old$ November 1st, 2009, 11:22 PM

can you give an example of non lenear equation?

Prove It · #6 $Old$ November 1st, 2009, 11:25 PM

Quote:

Originally Posted by transgalactic $View Post$

can you give an example of non lenear equation?

$\frac{d^2y}{dx^2} + 3x\sin{y}\frac{dy}{dx} + 6y = 0$ .

Can you see how the coefficient of $\frac{dy}{dx}$ is "more than" a function of $x$ ? It has a function of $y$ in it as well.

transgalactic · #7 $Old$ November 1st, 2009, 11:28 PM

yes i understand thanks