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ari10184
21.09.2019 •
Mathematics
Given y'' - x^2y' + xy = 0 about (0,0)
a. find two solutions of the given differential equation.
b. solve the ivp if y(0) = 2, and y'(0) = 5
i am having trouble with the re-indexing and solving for ck's
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Ответ:
Differentiating gives
Substituting into the ODE yields
By extracting the first term (
Next, since the first series starts with a constant term while the other two start with a quadratic term, remove the first two terms:
Now shift the index so that the series begins at the same starting index by replacing
So now the series can be joined into one:
which means the ODE is equivalent to
It follows that
with the initial values
Because
Also, because the RHS of the recurrence vanishes when
So the hard part is finding a solution for
and so on, following a general pattern of
valid for
So the series solution is
So the two solutions are
To solve the IVP, notice that when
so the solution to the IVP is the same as that to the general solution with the unknowns replaced accordingly.
Ответ:
:)))
Step-by-step explanation: