Answer to Question #222904 in Differential Equations for Anees

Question

x^2y''-7xy'-20y=0 if y1=x^10

Answer 1

Solution

Once we have this first solution we will then assume that a second solution will have the form

y₂(x) = v(x)y₁(x) or y₂(x) = v(x)x¹⁰    

Differentiating

y₂‘(x) = v’(x)x¹⁰ + 10v(x)x⁹ , y₂‘’(x) = v’’(x)x¹⁰ + 20v’(x)x⁹ + 90v(x)x⁸    

Plugging these into the differential equation gives

v’’(x)x¹² + 20v’(x)x¹¹ + 90v(x)x¹⁰ - 7v’(x)x¹¹ - 70v(x)x¹⁰ - 20v(x)x¹⁰  = 0   

v’’(x)x + 13v’(x) = 0   

Change of variable w(x) = v’(x), v’’(x) = w’(x)  =>  xw’ + 13 w = 0  => w(x) = Cx^-13  =>  v(x) = -Cx^-12/12+ D

Therefore   for C = -12 , D = 0 we obtain y₂(x) = v(x)x¹⁰   = x^-2   

Then general solution will then be

y(x) = Ax¹⁰ + Bx^-2