The functions y2 x2 and y2 x4 are two solutions of the equ

The functions y_2 = x^2 and y_2 = x^4 are two solutions of the equation x^2y\" - 5ry\" + 8y = 0. Find the general solution of this equation.

Solution

x²y\'\'-5xy\'+ 8y =0

above equation is Cauchy\'s homogeneous Linear DE

whose general solution is in the form of y =c₁e^m₁logx + c₂e^m₂logx

^{given that} y = x² and y= x⁴ are the solution hence m1 = 2 and m2 = 4

y =c₁e^2logx + c₂e^4logx is the generalized solution