In Problems 1-10, use a substitution y=xr to find the general solution to the given equation for x>0.
The general solution for the given equation is y=c1x-1/2+√13/2 +c2x-1/2-√13/2 .
In mathematics, a Cauchy problem is one in which the solution to a partial differential equation must satisfy specific constraints specified on a hypersurface in the domain.
An initial value problem or a boundary value problem is both examples of Cauchy problems.
The equation will be in the form of ax2y"+bxy'+cy=0.
The given equation is,
Let L be the differential operator defined by the left-hand side of equation, that is
Substituting the w(r,x) in place of y(x), you get
=x2 (r(r-1)) xr-2+2x (r) xr-1-3 (xr)
Solving the indicial equation
There are two distinct roots,
r1= -1/2+√13/2 and r2= -1/2-√13/2
Thus there are two linearly independent solutions given by,
y1=c1x-1/2+√13/2 and y2=c2x-1/2-√13/2
Hence, the general solution for the given equation will be,
Show that fn(0)=0 for n=0,1,2.... and hence that the Maclaurin series for f(x) is 0+0+0+.... , which converges for all x but is equal to f(x) only when x=0 . This is an example of a function possessing derivatives of all orders (at x0 =0 ), whose Taylor series converges, but the Taylor series (about x0 =0) does not converge to the original function! Consequently, this function is not analytic at x=0.
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