Recall from Chapter 3, equation (8.5), that a set of functions is linearly independent if their Wronskian is not identically zero. Calculate the Wronskian of each of the following sets to show that in each case they are linearly independent. For each set, write the differential equation of which they are solutions. Also note that each set of functions is a set of basis functions for a linear vector space (see Chapter 3, Section 14, Example 2) and that the general solution of the differential equation gives all vectors of the vector space.

$1,x,x^2$

Given equation is$1,x,x^2$

Definition of Wronskian:

A mathematical determinant whose first row consists of $n$functions of $x$and whose following rows consist of the successive derivatives of these same functions with respect to role="math" localid="1664294588694" $x$.

First of all, it is good to notice that there are three solution which means that the differential equation is third order. To prove that the two solutions are linearly independent we need to find the Wronskian, and if it was no identically zero, that they are independent

$$\begin{gathered} W=\left|\begin{array}{ccc}1& x& x^2\\ 0& 1& 2x\\ 0& 0& 2\end{array}\right| \\ =2 \end{gathered}$$

Which means our solution are linearly independent.

Therefore, the general solution is

$y=1+x+x^2$

Now, since it is sure that we have a differential equation then need first to write it in general$a_3{y}^{\text{'}\text{'}\text{'}}+a_2{y}^{\text{'}\text{'}}+a_1{y}^{\text{'}}+a_{0}y=0$.

To find the differential equation it is differentiate the general solution three times

$$\begin{gathered} y^{\text{'}}=1+2x \\ {y}^{\text{'}\text{'}}=2 \\ {y}^{\text{'}\text{'}\text{'}}=0 \end{gathered}$$

therefore, the differential equation for the solution is $y\text{'}\text{'}\text{'}=0$.