What is the main difference between solutions of linear ODEs

What is the main difference between solutions of linear ODE\'s and nonlinear ODE\'s assuming the linear/non linear ODE satisfy the conditions for existence - uniqueness of theorems 2.4.1 and 2.4.2 Theorem 2.4.1: If the function p & g are continuous on an open interval I: d

Solution

If the equation would have had ln(y)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+by=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 xx\'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 yy 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. dydxdydx; yes. dydx4dydx4, no.

Each coefficient depends only on the independent variable xx.

yyyy makes it nonlinear as has been said, because that coefficient on yy is not in xx. Had that coefficient been a constant, you would have been correct to call it linear, since constants can be functions of xx. Like, f(3)=xf(3)=x. Its graph is a line, i.e. linear function.