Indicate if the following systems are lineare or non linjear systems d^2x/dt+5dx/dt+10x = 0

You have only provided one Differential Equation (DE), it looks like you intended listing more.

The equation you wrote contains an incorrect d²x/dt, it is likely to be d²x/dt² + 5dx/dt + 10x = 0, which is linear. Unless it is (dx/dt)² + 5dx/dt + 10x = 0, then it is nonlinear.

Not to worry though, I will explain what linear and nonlinear DE's are.

Step-by-step explanation:

LINEAR DE: This is the kind of DE in which the functions of the dependent variable are linear. There are no powers of the dependent variable and/or its derivatives, there are no products of the dependent variable and its derivative, there are no functions of the dependent variable like cos, sin, exp, etc.

Example:

* 5d²x/dt² + dx/dt - x = 2t

This is linear, as it satisfies all the conditions.

NONLINEAR DE: If any condition explained for linear DE is not satisfied, then it is called nonlinear.

Example:

* d²x/dt² - sinx = 0

This is nonlinear because of the presence of sinx.

* d²x/dt² + xdx/dt = 0

This is nonlinear because of the product of the dependent variable, x, and its derivative, dx/dt.

* d²x/dt² + x² = 0

This is nonlinear because a function of the dependent variable is not linear. You shouldn't have x².

* (dx/dt)³ + 3dx/dt = 0 is equally nonlinear. You can't have nonlinear functions of the dependent variable or its derivatives.

I hope this helps answer the remaining parts of your question.