Question

4) (10 pts.) Solve using the method of reduction of order (not the formula). Show all steps. yi-r2 is one solution. xzy"-3ry' + 4y-0,

Answer 1

x^2 y Prime - 3xy prime + 4y = 0 y_1 = x^2 let y_2 = v(x) middot y_1(x) = v(x) middot x^2 y prime_2 = V 2x + x^2 v prime y Prime_2 = V 2 + 2x middot v prime + x^2 v Prime + v prime middot 2x = x^2 v Prime + 4x v prime + 2v Therefore x^2 y Prime_2 - 3xy prime_2 + 4y_2 = 0 = > x^2 (x^2 v Prime + 4x v prime + 2v) - 3x (2xv + x^2 v prime) + 4x^2 v = 0 = > x^4 v Prime + 4 x^3 v prime + 2 x^2 v - 6 x^2 v - 3x^3 v prime + 4x^2 v = 0 = > x^4 v Prime + x^3 v prime = 0 = > x v Prime + v prime = 0 let w = v prime, w prime = v Prime = > x w prime + w = 0 = > w(x) = c_1/x
y(x)=c₁x²+c₂x²ln(|x|)