Linear Dependence of Functions with Absolute Value

In summary, the question is asking whether the functions f(x) = x^3 and g(x) = x^2|x| are linearly dependent or independent. After calculating the derivatives of both functions, it is determined that they are linearly independent due to the fact that they cannot be multiplied by a constant to become the same function for all values of x. The absolute value plays a role in this determination.
  • #1
jinksys
123
0

Homework Statement


Determine whether the pairs of functions are linear dependent or linearly independent.

f(x) = x^3
g(x) = x2|x|

Homework Equations





The Attempt at a Solution



g(x)=x^2|x| = x^2*sqrt[x^2] = sqrt[x^6] = x^3

f'=3x^2
g'=3x^2

fg'-f'g = 0

Linearly Dependent according to me, Linearly independent according to the book.

I assume it has to do with the absolute value, could someone enlighten me?
 
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  • #2
Should g'(x) be 2x|x|+x^2(|x|/x) ?
 
  • #3
note that f is positive for some values of x and negative for others whereas g is always positive. if these two were linearly dependent one would be a constant multiple of the other FOR ALL VALUES OF x. can we multiply a fully positive function by a number so that part of it becomes negative? no. therefore no linear dependence. note also this depends on the set over which g and f are defined.
 
  • #4
why are you take the derivative?
 

Related to Linear Dependence of Functions with Absolute Value

1. What is a linearly dependent differential equation?

A linearly dependent differential equation is one in which the dependent variable and its derivatives can be expressed as a linear combination of each other.

2. How can I determine if a differential equation is linearly dependent?

You can determine if a differential equation is linearly dependent by checking if the determinant of the coefficient matrix is equal to 0. If the determinant is 0, the equation is linearly dependent.

3. What is the significance of a linearly dependent differential equation?

A linearly dependent differential equation can provide useful information about the relationship between the dependent variable and its derivatives. It can also be used to solve for the dependent variable in terms of its derivatives.

4. Can a linearly dependent differential equation have multiple solutions?

Yes, a linearly dependent differential equation can have multiple solutions. This is because the equation is not uniquely determined by its initial conditions.

5. How are linearly dependent differential equations used in real-world applications?

Linearly dependent differential equations are commonly used in physics, engineering, and other scientific fields to model systems that exhibit linear relationships between variables. They are also used for predicting trends and behaviors in various systems.

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