In Problems 1-10, use a substitution y=xr to find the general solution to the given equation for x>0.
The general solution to the given equation x2y"(x)+6xy'(x)+6y(x)=0 is y=c1x-2+c2x-3.
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.
Let L be the differential operator defined by the left hand side of the equation.
By substituting you get,
=(r2-r) xr +6rxr +6xr
Solving the indicial equation.
The two distinct roots are,
There are two linearly independent solutions.
The general solution is y=c1x-2+c2x-3.
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