solve xd^2y/dx^2+y=0
by frobenius method
There is a nice discussion and a good example here:
http://en.wikipedia.org/wiki/Frobenius_method
where do you get stuck?
I know the steps, but I want to make sure the solution of the general solution
With thanks and appreciation to answer
To solve the differential equation using the Frobenius method, we assume a power series solution of the form y = ∑(n=0 to ∞) a_nx^(n+r), where r is a constant to be determined and a_n are the coefficients of the series.
Now, let's differentiate y with respect to x:
dy/dx = ∑(n=0 to ∞) a_n(n+r)x^(n+r-1)
Next, differentiate dy/dx with respect to x:
d^2y/dx^2 = ∑(n=0 to ∞) a_n(n+r)(n+r-1)x^(n+r-2)
We substitute these expressions into the original differential equation:
∑(n=0 to ∞) a_n(n+r)(n+r-1)x^(n+r-2) + ∑(n=0 to ∞) a_nx^(n+r) = 0
Now, we simplify the equation by shifting the index of summation in the first term:
∑(n=2 to ∞) a_(n-2)(n+r)(n+r-1)x^(n+r-2) + ∑(n=0 to ∞) a_nx^(n+r) = 0
Now, we need to equate the coefficients of like powers of x to obtain a recurrence relation. Equating the coefficients of x^(n+r-2) gives us:
a_(n-2)(n+r)(n+r-1) + a_n = 0
Now, we solve this recurrence relation to find the values of a_n in terms of the initial conditions. Since the equation is quadratic, we can solve for a_n in terms of a_(n-2):
a_n = -a_(n-2) / (n+r)(n+r-1)
Now, we need to determine the values of r that satisfy the recurrence relation. For a power series solution, we want to find values of r for which a_n does not become infinite for any value of n and a_n eventually becomes zero. This means that the recurrence relation must terminate. To achieve this, the coefficient of a_n in the recurrence relation must be zero. Thus, we set:
(n+r)(n+r-1) = 0
Solving this quadratic equation gives us two roots:
r = 0 and r = -1
Taking r = 0, we have:
a_n = -a_(n-2) / n(n-1)
Similarly, for r = -1, we have:
a_n = -a_(n-2) / (n+1)n
These recurrence relations will help us to determine the coefficients a_n. We start with the initial conditions (given in the problem) to find the specific values of a_0 and a_1. Then, we can use the recurrence relations to find the remaining coefficients a_2, a_3, and so on.
Finally, we substitute the values of the coefficients a_n back into the power series solution for y to obtain the general solution of the given differential equation.