Linearly Independent/Dependent

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SUMMARY

The discussion centers on the classification of linear and nonlinear differential equations, specifically focusing on the equation y' + P(x)y = Q(x). It is established that the linearity of such equations is determined by the degree of y and its derivatives, which must be first degree for the equation to be classified as linear. The presence of nonlinear functions, such as y^2 or y * ln(y), in the equation indicates nonlinearity. Therefore, equations like y' - y = xy^2 and y' - 2xy = (1/x)y * ln(y) are confirmed to be nonlinear.

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  • Understanding of first-order differential equations
  • Knowledge of linear and nonlinear functions
  • Familiarity with the concept of linear independence in solutions
  • Basic calculus, including derivatives and their properties
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JosephK
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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.
 
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JosephK said:
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.

What about y'-y=xy^2?

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

What about y'-2xy=1/x*y*lny?

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

Thank you.
 
JosephK said:
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.
What about y'-y=xy^2?

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.
What about y'-2xy=1/x*y*lny?

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:
JosephK said:
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.
JosephK said:
What about y'-y=xy^2?

Is this linear differential equation non-linear because the y to the right is not in the first degree?
No. This differential equation is nonlinear for the reason you say. A differential equation is either linear or nonlinear. It makes no sense to ask if a linear DE is nonlinear. If it's nonlinear it is not a linear differential equation.
JosephK said:
What about y'-2xy=1/x*y*lny?

Is this linear differential equation not linear because the coefficient of the y to the right hand side depends on y?
Your question should be, "Is this differential equation nonlinear ..." It is nonlinear because of the lny factor on the right side.
 

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