CAUCHY-EULER EQUATIONS ASSIGNMENT Solve the following Cauchy-Euler equations. Determine the general solution using variation of parameters Solution ax^2y\'\' + bxy\'+ cy = 0; this is called homogeneous Euler-Cauchy equation,its characteristic equation is am^2 + (b ??-a)m + c = 0: 1) x^2y\'\' +8xy\' +6y = 0 characteristic equation m^2 +7m+6 = 0 , m= -6,-1, hence y = C1*x^-6 + C2*x^-1. 2) x^2y\'\' +6xy\' + 4y =0 characteristic equation m^2 + 5m +4 = 0, m= -4,-1 ,hence y = C1*x^-4 + C2*x^-1 3) x^2y\'\' - 3xy\' +4y = 0 characteristics equation m^2-4m +4 = 0 , m = 2,2, hence y = (C1 + C2 ln x)*x^m = (C1 +C2lnx)*x^2 4) x^2y\'\' -5xy\' +8y = 0 characteristics equation m^2, - 6m +8 = 0 , m= 4,2 hence y = C1x^4 + C2x^2, solving y(2) = 32 & y\'(2) = 0, we get C1= -2 and C2 = 16, so y = -2x^4 + 16x^2. 5).

1. CAUCHY-EULER EQUATIONS ASSIGNMENT Solve the following Cauchy-Euler equations.
Determine the general solution using variation of parameters
Solution
ax^2y'' + bxy'+ cy = 0; this is called homogeneous Euler-Cauchy equation,its characteristic
equation is
am^2 + (b ??-a)m + c = 0:
1) x^2y'' +8xy' +6y = 0
characteristic equation m^2 +7m+6 = 0 , m= -6,-1, hence y = C1*x^-6 + C2*x^-1.
2) x^2y'' +6xy' + 4y =0
characteristic equation m^2 + 5m +4 = 0, m= -4,-1 ,hence y = C1*x^-4 + C2*x^-1
3) x^2y'' - 3xy' +4y = 0
characteristics equation m^2-4m +4 = 0 , m = 2,2, hence y = (C1 + C2 ln x)*x^m = (C1
+C2lnx)*x^2
4) x^2y'' -5xy' +8y = 0
characteristics equation m^2, - 6m +8 = 0 , m= 4,2 hence y = C1x^4 + C2x^2, solving y(2) = 32
& y'(2) = 0,
we get C1= -2 and C2 = 16, so y = -2x^4 + 16x^2.
5)