In differential equations how do you know whether a differential equation is lin
ID: 3199118 • Letter: I
Question
In differential equations how do you know whether a differential equation is linear or nonlinear? Do you care only about the highest derivative of the dependent variable or do you also care if there is a dependent variable to a power?How do you know the order of the differential equation? In differential equations how do you know whether a differential equation is linear or nonlinear? Do you care only about the highest derivative of the dependent variable or do you also care if there is a dependent variable to a power?
How do you know the order of the differential equation?
How do you know the order of the differential equation?
Explanation / Answer
Linear vs. Non-linear
An ODE for y = y(t) is linear if it can be written in the form
an(t) yn+ an-1(t) yn-1 + · · · + a3(t) y3 + a2(t) y'' + a1(t) y'+ a0(t) y = g(t)
where the ‘coefficient’ functions g(t), a0(t), ..., an(t) can be any functions of t, (including the zero function),
Example1. y''+ 2t3 y' = 5ty + t2 is a second order linear ODE, since it can be written
y''+ 2t3 y' + (-5ty) = t2
Ex2. y''= yy' ? 9 is a second order nonlinear ODE, due to the term yy'
Ex3. y2 + y' = ? t is a first order nonlinear ODE
Note- We care only power of dependent variable for example
y -> linear, y2 -> non linear
Order of differential Equation - Order of differential is the highest derivative of dependent variable
Example-y''+ 2t3 y' = 5ty + t2
It is 2nd order differential euation
Example- y2 + y' = ? t
It is a first order Differential equation
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