# Linearly Independent/Dependent

Is a linear equation y'+P(x)y=Q(x) not linear if P(x) and Q(x) are not linearly dependent function?

Does linearly dependent mean a constant multiplied by P(x) will equal Q(x)?

Thank you.

Related Calculus and Beyond Homework Help News on Phys.org
LCKurtz
Homework Helper
Gold Member
Is a linear equation y'+P(x)y=Q(x) not linear if P(x) and Q(x) are not linearly dependent function?

Does linearly dependent mean a constant multiplied by P(x) will equal Q(x)?

Thank you.
No.

The equation itself may or not be linear. In this case, it is because y and y' occur only as first degree. It has nothing to do with x.

An nth order linear homogeneous DE will have n linearly independent solutions. That is not the same concept. A first order DE as in your example can not have two linearly independent solutions. So if your example was y' + P(x)y = 0, and y is a solution, then any constant time y is a solution. Such solutions are linearly dependent.

In recognizing linear differential equations for example y'+3x^2y=x^2 I do not say this linear differential equation is linear because I can multiply x^2 by 3).

I should say because y and y' are of the first degree this equation is linear.

Is this linear differential equation non-linear because the y to the right is not in the first degree?

Is this linear differential equation not linear because the coefficient of the y to the right hand side depends on y?

Thank you.

In recognizing linear differential equations for example y'+3x^2y=x^2 I do not say this linear differential equation is linear because I can multiply x^2 by 3).

I should say because y and y' are of the first degree this equation is linear.
Correct.

Is this linear differential equation non-linear because the y to the right is not in the first degree?
Correct.
edit: I read this too fast. As Mark44 points out below, an equation is either linear or not. This equation is non-linear.

Is this linear differential equation not linear because the coefficient of the y to the right hand side depends on y?
It's non-linear because a non-linear function of y (y * ln y) appears in the equation.

Last edited:
Mark44
Mentor
In recognizing linear differential equations for example y'+3x^2y=x^2 I do not say this linear differential equation is linear because I can multiply x^2 by 3).

I should say because y and y' are of the first degree this equation is linear.
Yes. This is a linear differential equation. A linear DE is one in which y, y', y'', etc. occur to the first power. How the independent variable (x in this case) occurs doesn't enter into the description.