Consider the differential equation x2y′′ − 9xy′ + 24y = 0; x4, x6, (0, [infinity]). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x4, x6) = ≠ 0 for 0 < x < [infinity]. Form the general solution. y =

The functions satisfy the differential equation and linearly independent since W(x)≠0

Therefore the general solution is

Step-by-step explanation:

Given equation is

This Euler Cauchy type differential equation.

So, we can let

Differentiate with respect to x

Again differentiate with respect to x

Putting the value of y, y' and y'' in the differential equation

⇒m²-10m +24=0

⇒m²-6m -4m+24=0

⇒m(m-6)-4(m-6)=0

⇒(m-6)(m-4)=0

⇒m = 6,4

Therefore the auxiliary equation has two distinct and unequal root.

The general solution of this equation is

and

First we compute the Wronskian

         =x⁴×6x⁵- x⁶×4x³    

        =6x⁹-4x⁹

        =2x⁹

       ≠0

The functions satisfy the differential equation and linearly independent since W(x)≠0

Therefore the general solution is

the formula of a slope:

from a table we have

substitute:

the slope of f(x) is -1.

the graph of funcfion g(x) is a horizontal line. the slope of a horizontal line is equal 0. therefore the slope of g(x) is 0.

-1 < 0

b. x + 1

step-by-step explanation:

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