Determine the general solution of Euler eq that ia valid in any interal not including the singular point. x^2y'' +4xy' + 2y =0 Answer is y=c1(x^-1) + c2(x^-2)

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You need to assume that the general form of solution to differential equation is `y = x^r` .

Differentiating y with respect to x yields:

`y' = rx^(r-1)`

`y'' = r(r-1)x^(r-2)`

Substituting y'' and y' in equation yields:

`x^2*r(r-1)x^(r-2) + 4x*rx^(r-1) + 2x^r = 0`

You need to factor out `x^r`  such that:

`x^r(r(r - 1) + 4r + 2) = 0`

Dividing by `x^r`  yields:

`r^2 - r + 4r + 2 = 0`

`r^2 + 3r + 2 = 0`

You need to find the roots of quadratic equation such that:

`r^2 + 2r + r + 2 = 0`

`r(r+1) + 2(r+1) = 0`

Factoring out r+1 yields:

`(r+1)(r+2) = 0`

You need to solve equations r + 1 = 0 and r + 2 = 0 such that:

`r + 1 = 0 =gt r_1 = -1`

`r + 2 = 0 =gt r_2 = -2`

Hence, evaluating the general solution to equation yields `y = c_1*x^(-1) + c_2*x^(-2).`

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