The second order linear differential equations t²y" + bty + cy=0 are called Euler's equation. For this type of differential equations, we look for solutions of the type y(t) = t". a) Verify that y(t) = t" is a solution of t²y" + bty' + cy = 0 if r is a solution of ²+(b-1)r+c=0. b) If r²+ (b- 1)r + c = 0 has two distinct real roots then there are two fundamental solutions. In this case, write then and the general solution. c) If r² + (b − 1)r + c = 0 has two complex roots. Use the relations t" = erint and ea+iß = ea(cos ß+isin 3) to obtain two real solutions (fundamental set) and the general solution. d) If r² + (b-1)r + c = 0 has one real repeated root r₁ then y₁ (t) = t" is a solution. Apply reduction of order method to obtain another solution 2(t) that will constitute

The point x = 0 is a regular singular point of the differential equation. xy'' + y' + 15y = 0 Use the general form of the indicial equation (14) in Section 6.3 r(r − 1) + a0 r + b0 = 0    (14) to find the indicial roots of the singularity. (List the indicial roots below as a comma-separated list.) r =        Without solving, discuss the number of series solutions you would expect to find using the method of Frobenius. (a)Since these do not differ by an integer we expect to find one series solution using the method of Frobenius. (b)Since these are equal we expect to find two series solutions using the method of Frobenius.     (c)Since these are equal we expect to find one series solution using the method of Frobenius. (d)Since these differ by an integer we expect to find one series solution using the method of Frobenius. (e)Since these differ by an integer we expect to find two series solutions using the method of Frobenius.

1. Find a system of first-order differential equations that correspond to the given second order differential equation: t²u" + 3tu' +u = t² + 2 Show all steps, including which new variables you are assigning to which terms.