How to distinguish linear differential equations from nonlinear ones? I know, that e.g.: $$ y''-2y = \ln(x) $$ is linear, but $$ 3+ yy'= x - y $$ is nonlinear. Why?
5 Answers
Linear differential equations are those which can be reduced to the form $Ly = f$, where $L$ is some linear operator.
Your first case is indeed linear, since it can be written as:
$$\left(\frac{d^2}{dx^2} - 2\right)y = \ln(x)$$
While the second one is not. To see this first we regroup all $y$ to one side:
$$y(y'+1) = x - 3$$
then we simply notice that the operator $y\mapsto g(y) = y(y'+1)$ is not linear (for example we can take two functions $y_1$ and $y_2$ and notice that $g(y_1+y_2)\neq g(y_1) + g(y_2)$).
If the equation would have had $\ln (y)$ on the right, that also would have made it non-linear, since natural logs are non-linear functions. Remember that this has its roots in linear algebra: $y=mx+b$. You can analyse functions term-by-term to determine if they are linear, if that helps. The first time a term is non-linear, then the entire equation is non-linear.
Remember that the $x$s can pretty much do or appear however they want, since they're independent. Which means if you can't tell just by glancing, try to group all your $y$ terms to one side and then analyse them. Makes it much easier.
See, I was also overthinking this, but realised you have to go back to those definitions we're given.
Two criteria for linearity:
The dependent variable y and its derivatives are of first degree; the power of each y is 1. $\frac{dy}{dx}$; yes. $(\frac{dy}{dx})^4$, no.
Each coefficient depends only on the independent variable $x$.
$yy'$ makes it nonlinear as has been said, because that coefficient on $y'$ is not in $x$. Had that coefficient been a constant, you would have been correct to call it linear, since constants can be functions of $x$. Like, $f(3)=x$. Its graph is a line, i.e. linear function.
Always go back to the definitions. :-)
One could define a linear differential equation as one in which linear combinations of its solutions are also solutions.
Linear Differential equations are those in which the dependent variable and its derivatives appear only in first degree and not multiplied together
Because highest order derivative is multiplied with dependent variable $y$. Like $y y'$.
