Find the recurrence formula for the series solution of yt

Question

Series Solution of Differential Equations-

1. Show by inspection that N! is larger than x^N, as N tends to ∞, for any value of x.

2. For the function f(x) = exp(-2x):

a. Verify that a series expansion is possible using Taylor's expansion at x = 0.

b. Derive an expression for f(x) as a series expansion.

c. If only the first three terms of the series are used, calculate the error (in %) for x=1, using direct substitution in series terms.

d. Repeat 2c, but use the remainder term to find the error instead of substitution in the series terms.

3. For the ordinary differential equation (ODE):

d²y/dt² +  (t-1) dy/dx + (2t-3)y = 0

a. Find the recurrence formula for the series solution of y(t) around t=0.

b. Expand the solution near t=0 for the first powers of tⁿ, where n ≤ 3.

4. Find a series solution the Airy Differential equation using the same steps in Q3:

y'' + xy = 0

5. Check if t=0 is an ordinary or singular point for the ODE in problems 3, and x=0 for problem 4.

6. Check for the following ODE if they have ordinary, singular, or regular singular points at x=0:

a. 2x²y'' + 7x(x+1)y' - 3y = 0

b. x³y'' + 2x²y' + y = 0

7. Apply Frobenius Method to find one series solution for the following DE:

x²y'' + (x² - 2x)y' +2y = 0

8. Apply Frobenius Method to find a solution for the following Bessel's DE:

x²y'' + xy' + x²y = 0

9. Apply Frobenius Method to find a solution for the following Bessel's DE:

x²y'' + xy' + (x² - 4)y = 0

10. Evaluate the solutions for both problems 8 and 9 at x=0, 2, 4, 6, 8, 10, then plot them qualitatively (free hand plot) using these points.