Solved by Luca Imponenti
Find , for such that:
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for in with the initial conditions found.
Plot for for in .
Homogeneous Solution
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The homogeneous case is shown below:
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This equation has the following roots:
Which gives yields the homogeneous solution
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General Solution, n=4
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Using the taylor series approximation from earlier with we have
We know the particular solution, , ve will have this form:
taking the derivatives of this solution
and
Plugging the above equations into the original ODE yields the following matrix equation:
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The unknown vector can be easily solved by forward substitution,the following values were calculated in matlab:
So the particular solution is
We can now find the general solution for n=4, .
Solving using the initial conditions yields;
General Solution, n=7
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Using the taylor series approximation from earlier with we have
In a similar fashion we construct a matrix equation for n=7:
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Solving:
So the particular solution is
We can now find the general solution for n=7, .
Solving using our initial conditions yields
General Solution, n=11
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Using the taylor series approximation from earlier with we have
Finally, we write out the matrix equation for n=11:
Solving the system in matlab:
So the particular solution is
We can now find the general solution for n=11, .
Solving using our initial conditions yields
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