Question

Is the given function phi(x) = x^2 - x^-1 an explicit solution to the linear equation d^2y/dx^2 - 2/x^2 y = 0? Circle your answer. (a) yes (b) no

Answer 1

Yes

Explanation:

We are given that a function
`\phi(x)=x^2-x^(-1)`

We have to find that given function is an explicit solution to the linear equation

`(d^2y)/(dx^2)-(2)/(x^2)y=0`

If given function is an explicit solution of given linear equation then it satisfied the given differential equation

Differentiate w.r.t x

Then we get
`\phi'(x)=2x+x^(-2)`

Again differentiate w.r.t x

Then we get

`\phi''(x)=2-(2)/(x^3)`

Substitute all values in the given differential equation

`2-(2)/(x^3)-(2)/(x^2)(x^2-x^(-1))`

=
`2-(2)/(x^3)-2+(2)/(x^3)=0`

Hence, given function is an explicit solution of given differential equation.

Therefore, answer is yes.