Differential Equations? Should be easy.?

Question

How do you tell the difference between a linear differential equation and a nonlinear one? Thanks a lot.

Answer 1

It's linear if the function (say x) appears only in terms by itself or multiplied by a constant, and it's not raised to any powers.

So x' + 3 = 5x'' is linear because x was never raised to a power and it only appeared by itself or with a constant.

x'^2 + 4 = 4x is not linear because one of the terms has the x squared.

sin(x') + 5 = 7x + x'' is not linear because of the sin(x) term.

5x' + 1 = 7t^2 + t*sin(t) is linear though, because like the first examples, the x only appears with a constant.

Answer 2

The way I think of it is that the dependent variable and all of its derivatives should be linear. For example, if y is a function of x, then y should appear in its first degree, no y^2 or y^-4, same thing for the derivatives. There should be no (y'')^4 and then none of the derivatives and y should be multiplied together like y*y'=x or something. If the dependent variable and all of its derivatives are linear (wherever they appear), the differential equation is linear.

Remember that the restriction is on the dependent variable. The independent variable could do anything like x*y'=y or x^2+y'=0 are valid.

Answer 3

yepp.. you just have to solve them methodically. there are a few tips you can use while solving them.. look them up online.. they are extremely lengthy though. they might look difficult but once you know the formulas you could use, it's extremely easy. a bit of practice is all it takes. and yes i solved them by hand. in india we don't yet use MATLAB and other programs to do calculus..

Answer 4

Linear diff eq - all the derivatives are terms like terms d^n f(x)/dx^d where d^n /dx^n is the nth derivative with respect to x

e.g.
a(x)*d^nf (x)/dx^n +b(x)*d^(n-1) f(x)/dx^(n-1) + ...+ z(x) f(x) = g(x)

Nonlinear diff eq. has derivatives raised to powers. e.g.

d^2 f(x)/dx^2 + b(x)*(d f(x)/dx)^2+ f(x) = g(x)

Note that (d f(x)/dx)^2 = df(x)/dx * d(f(x)/dx