solve (x-1) (d^2 y)/〖dx〗^2 +2dy/dx=0 using series method
(x-1)y" + 2y' = 0
If we expand about x=0, we assume
y = ∑anx^n
y' = ∑nanx^(n-1)
y" = ∑n(n-1)nx^(n-2)
Plug these into our DE and we have
(x-1)∑n(n-1)nx^(n-2) + 2∑nanx^(n-1) = 0
∑n(n-1)nx^(n-1) - ∑n(n-1)nx^(n-2) + 2∑nanx^(n-1) = 0
∑n(n-1)nx^(n-2) + ∑(2n+n-1)anx^(n-1) = 0
∑(n+1)(n+2)n+2x^n
+ ∑(3n+2)an+1x^n
∑((n+1)(n+2)n+2+(3n+2)an+1)x^n = 0
So, now we have the recurrence relation
(n+1)(n+2)n+2+(3n+2)an+1 = 0
I expect you can take it from here; if you get stuck, read the excellent article at
http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx
To solve the given second-order differential equation using the series method, we need to express the unknown function y as a power series and substitute it into the equation. This method is also known as the Frobenius method.
Step 1: Assume a Power Series Solution
Let's assume that the solution can be represented as a power series of x, i.e., y = ∑(n=0 to ∞) aₙx^(n + r), where aₙ represents the coefficients of the series, r is a constant, and x^(n + r) denotes the term (x raised to the power of (n + r)).
Step 2: Compute the Derivatives
Next, we need to calculate the derivatives of y with respect to x. We'll need the first and second derivatives for the given equation.
- First Derivative: dy/dx = ∑(n=0 to ∞) aₙ(n + r)x^(n + r - 1)
- Second Derivative: (d²y)/dx² = ∑(n=0 to ∞) aₙ(n + r)(n + r - 1)x^(n + r - 2)
Step 3: Substitute into the Original Equation
Now, substitute the assumed power series solution and its derivatives into the given differential equation.
(x - 1) ∑(n=0 to ∞) aₙ(n + r)(n + r - 1)x^(n + r - 2) + 2 ∑(n=0 to ∞) aₙ(n + r)x^(n + r - 1) = 0
Step 4: Simplify the Equation
Now, simplify the equation by rearranging and collecting like terms:
∑(n=0 to ∞) aₙ(n + r)(n + r - 1)x^(n + r - 1) - ∑(n=0 to ∞) aₙ(n + r)(n + r - 1)x^(n + r - 2) + 2 ∑(n=0 to ∞) aₙ(n + r)x^(n + r - 1) = 0
Step 5: Determine the Value of r
To make the resulting equation solvable, we set the coefficient of the lowest power term (x^(n + r - 2)) to zero. This will give us an indicial equation to find the value of r.
For the equation to be balanced, we equate the coefficient of x^(n + r - 2) to zero:
aₙ(n + r)(n + r - 1) - aₙ(n + r)(n + r - 1) = 0
Simplifying the equation, the terms involving 'n' cancel out, and we obtain:
r(r - 1) = 0
Now, solve the above equation to find the values of r.
Step 6: Solve for the Coefficients aₙ
Once we have the value(s) of r, substitute it back into the original series expression for y.
y = ∑(n=0 to ∞) aₙx^(n + r)
Now, we can solve for the coefficients aₙ by plugging in the power series expression for y, with the values of r obtained.
By substituting the values of aₙ and r into the series expression, we obtain the final solution for y in terms of x.
Note: The specific values of r will determine the form of the power series, and the recurrence relation satisfied by the coefficients aₙ will depend on the value of r.