How to distinguish linear differential equations from nonlinear ones? I know, that e.g.:y″−2y=ln⁡(x)is linear, but3+yy′=x−yis nonlinear. Why?

Question

Daniella Cochran · Accepted Answer

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:(d2dx2−2)y=ln⁡(x)While the second one is not. To see this first we regroup all y to one side:y(y′+1)=x−3then we simply notice that the operator y↦g(y)=y(y′+1) is not linear (for example we can take two functions y1 and y2 and notice that g(y1+y2)≠g(y1)+g(y2)).

hercegvm · Accepted Answer

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 xs can pretty much do or appear however they want, since they&#039;re independent. Which means if you can&#039;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&#039;re given.Two criteria for linearity:1.The dependent variable y and its derivatives are of first degree; the power of each y is 1. dydx;2. Each coefficient depends only on the independent variable x.yy′ makes it nonlinear as has been said, because that coefficient on y′ is not 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.

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?

Open question

Answer & Explanation

New Questions in Calculus 2