Question

need som help plz (DE's) ! " We live in a world where joy and empathy and pleasure are all around us, there for the noticing. Ira Glass " any way , my qs is how to know if the ODE is linear or not ? plz give an example :'(

Answer 1

Say you have a function \(y\) that depends on \(x\). This function is linear if you can write it in the form
\[y=ax+b\]
where \(a,b\) are coefficients. In the context of precalculus, for example, these coefficients are generally taken to be constant numbers.

Now when you're considering a differential equation, you have a function, call it \(y'\), that depends on its antiderivative \(y\). Like with the form above, you would call an ODE linear if you can write it in the form
\[y'=ay+b\]
The difference here is that the coefficients don't necessarily have to be constants. For the most part, you'll likely encounter non-constant coefficients that are functions themselves of the independent variable(s), but not of \(y\) or any of its derivatives.

Some examples:
\[\begin{align*}
y'&=2y-5&\bf{linear}\\
x~y'&=x^2y+\frac{2}{x}&\bf{linear}\\
(\ln x)~ y'&=\sqrt{x}^{|x|}y+\sin(\cos e^x)&\bf{linear}\\
y~y'&=4y+1&\bf{nonlinear}\\
y'&=\frac{d}{dy}\left[\int_{-\infty}^yt!~e^{-t}~dt\right]&\bf{nonlinear}\\
(y')^x&=\frac{x}{y}-\pi^y\Gamma[xy]&\bf{nonlinear}
\end{align*}\]
Before I get carried away here... don't be discouraged by the complexity of some of these examples. The basic take-away is that you can't have powers of \(y\) or \(y'\), products of \(y\) and \(y'\), or functions of \(y\) that aren't (for lack of a better term) linear.

Also, notice how the coefficients of the ODE need not be linear themselves. They are not subject to the same conditions as \(y\) and \(y'\) because their linearity is of no concern to us.