Question

1 Solve by using power series: 2)-y = ex. Find the recurrence relation and compute the first 6 coefficients (a -as). Use the methods of chapter 3 to solve the differential equation and show your chapter 8 solution is equivalent to your chapter 3 solution.

Answer 1

We solved all the parts in detail. Please find the complete solution below.

Let y = sigma^infinity_n = 0 a_n x^2 be the solution of the 2y^derivative - y = e^x then y^derivative = sigma^infinity_n = 0 n a_n x^n -1 & e^x = sigma^infinity_n = 0 x^n/n! therefore 2y^derivative - y = e^x implies 2 sigma^infinity_n = 0 n a_n x^n -1 - sigma^infinity_n = 0 a_n x^n = sigma^infinity_n = 0 x^n/n! implies 2 sigma^infinity_n = 1 n a_n x^n -1 - sigma^infinity_n = 0 a_n x^n = sigma^infinity_n = 0 x^n/n! equation 1 Now we change the do many variables n by m + 1 in the first summation to get [n = m + 1 implies m = n -1] 2 sigma^infinity_n = 1 n a_n x^n -1 = 2 sigma^infinity_m = 0 (m + 1) a_m + 1 x^m again changing the do many variable m by n and from *, we get 2 sigma^infinity_n = 0 (n