Question

Solve the initial value problem 36x2y′′−182xy′+110y=0,y(1)=1,y′(1)=3.

(1) The largest intervals that may be considered for the domain of the solution to 36x2y′′−182xy′+110y=0 are either For −[infinity] type -inf and for [infinity] type inf.
(2) Let C1 and C2 be arbitrary constants. The general solution to the homogeneous differential equation 36x2y′′−182xy′+110y=0 is the function y(x)=C1y1(x)+C2y2(x)=C1+C2

Answer 1

The question involves solving a second-order linear homogeneous differential equation with variable coefficients to find the general solution and apply initial conditions to find specific constants.

Step-by-step explanation:

The initial value problem given is a second-order linear homogeneous differential equation with variable coefficients. The form of the differential equation is 36x2y'' - 182xy' + 110y = 0.

To solve this, we look for solutions of the form y = xr, where r is a number to be determined. Plugging this form into the differential equation yields a characteristic equation whose roots give us the values of r.

Using the given initial conditions, y(1)=1 and y'(1)=3, we can determine the arbitrary constants in the general solution.

Lastly, it is necessary to check our solution against the differential equation to ensure it satisfies both the equation and the initial conditions.