Question

Give an interval over which the set of two functions 𝒇_𝟏(𝒙)=𝒙^𝟐 and 𝒇_𝟐(𝒙)=𝒙|𝒙| is linearly independent. Then give an interval over which the set consisting of 𝒇_𝟏 and 𝒇_𝟐 is linearly dependent.

Short answer

The interval over which the set consisting of F1 and F2 is:

Linearly independent on the interval (−∞,∞).

Linearly dependent on the interval (−∞,0) or (0,∞).

Step-by-step solution

Define

According to Theorem 4.1.3: Criterion for linearly independent Solutions, the set of two solutions 𝑦1 and 𝑦2 is linearly independent on an interval 𝐼 if and only if 𝑊(𝑦_1,𝑦₂) ≠ 0 for every 𝑥 in the interval.

Solve the differential equations.

The function, 𝑓₂(𝑥)=𝑥|𝑥| can be broken down into two parts.

$f_2\left(x\right)=\left\{\begin{array}{l}-x^{2}for(-\infty ,0)\\ x^{2}for(0,\infty )\end{array}\right.$

Since 𝑓₂(𝑥) changes its sign at 𝑥=0 in the interval (−∞,∞), neither of functions 𝑓₁(𝑥)=𝑥²and 𝑓₂(𝑥)=𝑥|𝑥| can be written as a linear combination of the other, and therefore, we conclude that the set of functions is linearly independent on the interval (−∞,∞).

For the interval (−∞,0), the Wronskian is

$$\begin{gathered} 𝑊(𝑓1(𝑥),𝑓2(𝑥\left)\right)=𝑊(𝑥2,-𝑥2) \\ =\left|\begin{array}{cc}{x}^2& -x^2\\ 2x& -2x\end{array}\right| \\ =x^2\cdot (-2𝑥)-2𝑥\cdot (-{𝑥}^2) \\ =-2x^3+2x^3=0 \end{gathered}$$

Similarly, for the interval (0,∞), the Wronskian

𝑊(𝑓₁(𝑥), 𝑓₂(𝑥)) = 𝑊(𝑥², 𝑥²) = 0

For both the intervals, since 𝑊(𝑓₁₍𝑥), 𝑓₂(𝑥))=0 for all 𝑥, we conclude that the set of functions is linearly dependent on the interval (−∞,0) or (0,∞).

An interval over which the set consisting of 𝑓₁ and 𝑓₂ is linearly dependent is

Linearly independent on the interval (−∞,∞).

Linearly dependent on the interval (−∞,0) or (0,∞).