Question

show that the function 1, x and x^2 are linearly independent. Hence find the differential equation whose solution are 1, x and x^2.​

Answer 1

Answer:We are given that function  are the roots of differential equation

We have to show that   are linearly independent.

Linearly independent function: Those functions  in which any function is not a linear combination of other functions in given set of functions.

We are forming a matrix from the coefficient of given set of functions

$A=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$

Any row or column is not a linear combination of other two rows or columns.

Therefore, rank of given matrix=3

Hence,$1,x,x^{2}$  are linearly independent .

Step-by-step explanation: