How many perfect cubes are there between 2 to 200 ? ANSWER fast
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Use reduction of order. Given a solution , look for a second solution of the form
.
Compute the first two derivatives of :
Substitute them into the ODE:
Now substitute and you end up with a linear ODE:
Multiply through both sides by (if you're familiar with the integrating factor method, this is it):
Bear in mind that in order to do this, we require . Just to avoid having to deal with absolute values later, let's further assume
.
Notice that the left side is the derivative of a product,
Integrate both sides with respect to :
Solve for :
Solve for by integrating both sides:
Integrate by parts:
Solve for :
But since is already accounted for, the second solution is just
Still, the general solution would be
