Question

Having some trouble in my signals and systems course, How can I test a differential equation (DEQ) to determine if it is linear or not?
I've been given a list of rules by another chegger perhaps I'm misunderstanding them.

Here's the list of conditions a fellow chegger gave me:

The differential equation is linear if these conditions are satisfied:

1)the degree of the dependent variable is 1

2)the degree of differential equation is 1

3)dependent variable and its derivative are not multiplied together

4)transcendental function(sinx, cosx, lnx, exp(x)..)  should not be a function of dependent variable

Here's the actual Question that I need help with:

True / False , The following DEQ is Linear:

y''(t) + 22ty'(t) + 3y(t) = x'(t) + 5x(t +2)

The Solutions say that the system is linear and the answer is True, but doesn't the 22ty'(t) violate rule 3 making this a nonlinear DEQ?

An in depth explanation would be very much appreciated!

Answer 1

Given DEQ is linear .

you are saying that the term "22 t y'(t) " violating rule 3 . but , NO.

why because , here independent variable is " t " and dependent variable is " y " .

so , in " 22 t y'(t) " we are having independent variable multiplying with derivative of dependent variable . so rule 3 is also satisfying .

lets see , when will rule 3 violate ,

if you have a term like " 22 y y'(t) " , then rule 3 gets violated .

since this term have dependent variable and its derivative are multiplied together . hope you get the point.

please give upvote if you like my explanation . thank you.