Respuesta :
Answer:
m = 6,9
Step-by-step explanation:
We are given that [tex]y = x^m[/tex] is a solution to given differential equation.
[tex]x^2y'' - 14xy' + 54y = 0[/tex]
First we 3evaluate the value of:
[tex]y'' = m(m-1)x^{(m-2)}[/tex]
[tex]y' = mx^{(m-1)}[/tex]
Putting these value in the above differential equation, we get,
[tex]x^2m(m-1)x^{(m-2)} + 14xmx^{(m-1)} + 54x^m = 0[/tex]
[tex]x^m[m(m-1) -14m + 54] = 0[/tex]
[tex]x^m(m^2 - 15m + 54) = 0[/tex]
[tex](m^2 - 15m + 54) = 0[/tex]
[tex](m-9)(m-6) = 0[/tex]
[tex]m = 9,6[/tex]
Thus, for m = 9,6 , the function [tex]y = x^m[/tex] is a solution of given differential equation.
Values of m so that the function [tex]y = x^m[/tex] is a solution of the given differential equation are 6 and 9 and this can be determined by differentiating the [tex]y = x^m[/tex].
Given :
- [tex]y = x^m[/tex]
- [tex]x^2y"-14xy'+54y = 0[/tex] ------ (1)
The following calculations can be used to determine the value of 'm':
[tex]y = x^m[/tex] ---- (2)
Now, differentiate the above equation with respect to 'x'.
[tex]\dfrac{dy}{dx}=\dfrac{d}{dx}(x^m)[/tex]
[tex]y' = mx^{m-1}[/tex] ----- (3)
Now, differentiate equation (2) with respect to 'x'.
[tex]\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}(mx^{m-1})[/tex]
[tex]y" = m(m-1)x^{m-2}[/tex] ----- (4)
Now, put the values of y, y', and y" in equation (1).
[tex]x^2\times m(m-1)x^{m-2}-14x(mx^{m-1})+54x^m=0[/tex]
[tex]x^m(m(m-1)-14m+54)=0[/tex]
[tex]x^m(m^2-15m+54)=0[/tex]
(m - 9)(m - 6) = 0
m = 9 , 6
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https://brainly.com/question/13077606
