Question

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 basic 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.

$e^{-x},e^{-4x}$

Short answer

The Wronskian of the given set is $y^{\text{'}\text{'}}+5y^{\text{'}}+4y=0.$

Step-by-step solution

Given information

The vector $e^{-x},e^{-4x}$

The Wronskian

Józef Hoene-Wronski (1812) introduced the Wronskian (or Wronskian) determinant, which was named by Thomas Muir (1882, Chapter XVIII). It is sometimes used in the study of differential equations to show linear independence in a set of solutions.

Calculate the Wronskian of each of the following sets.

To prove that the two solutions are linearly independent find the Wronskian, and if it was not identically zero, they are independent

$$\begin{gathered} W=\left|\begin{array}{cc}{e}^{-x}& e^{-4x}\\ -e^{-x}& -4e^{-4x}\end{array}\right| \\ =-4e^{-5x}+e^{-5x} \\ =-3e^{-5x} \end{gathered}$$

It means our solutions are linearly independent. Again, the general solution is

$y=e^{-x}+e^{-4x}$

The general form of second-order differential equation with constant coefficients and zero right-hand side is given in the equation, that is,

$a_2{y}^{\text{'}\text{'}}+a_1{y}^{\text{'}}+a_{0}y=0$

If it compares with our solution, find that$c_1=c_2=1,a=-1$, and$b=-4$. As it knows, such a solution arises from an auxiliary equation that has two different real roots, therefore, the auxiliary equation is

$$\begin{gathered} (D+1)(D+4)y=\left(D^2+5D+4\right)y \\ =0 \end{gathered}$$

and the differential equation that has the general solution $y=e^{-x}+e^{-4x}$ is $y^{\text{'}\text{'}}+5y^{\text{'}}+4y=0$.