Question

Chapter 5, Section 5.2, Additional Question 01 Consider the following differential equation (10 2 y 20y 0, o = 0. (a) Seek a power series solution for the given differential equation about the given point a; find the recurrence relation Enclose numerators and denominators in parentheses. For example, (a - b)/ (1+n). Use an asterisk, *, to indicate multiplication. For example, 2* f(x), a* x* (b)* (c* x + d) b*tan (a* 0) or e(a**) *b a1+ a+2 an. (b) Find the first four terms each oftwo solutions y, and y, (unless the series terminates sooner). Write the first solution, y (x), using the even exponents for Fnter an exact answer. Use an asterisk, , to indicate multiplication. For example, 2* f(x), a* x* (x + b) * (cx+ d) b* tan (a* 0) or ea*a) * b +... +... (c) By evaluating the Wronskian W (y1,) (xo), show that h and y2 form a fundamental set of solutions. Use an asterisk, *, to indicate multiplication. For example, 2* f(x), a*x* (+b)* (c* x + d), b tan (a*e) or ela) *b W (,2) (o)

Answer 1

y'' + 9y = 3 e^-t^2, y (0) = y' (0) = 0 (10 + x^2) y'' - xy' + 20 y = 0, x_o = 0 rightarrow (1) Let y = sigma^infinity_n = 0 a_n x^n Implies y' = sigma^infinity_n = 1 n a_n x^n-1 Implies y'' = sigma^infinity_n = 2 n (n - 1) a_n x^n-2 Therefore (1) Implies sigma^infinity_n = 2 10 n (n - 1) a_n x^n-2 + sigma^infinity_n = 2 n (n - 1) a_n x^n - sigma^infinity_n = 1 n a_n x^n + sigma^infinity_n = 0 20 a_n x^n = 0 Implies sigma^infinity_n = 0 10 (n + 2) (n + 1) a_n+ 2 x^n + sigma^infinity_n = 2 n (n - 1) a_n x^n - sigma^infinity_n = 1 n a_n x^n + sigma^infinity_n = 0 20 a_n x^n = 0 Implies 10 (2) a_2 + 10 (3) (2) a_3 x - a_1 x + 20 a_0 + 20 a_1 x + sigma^infinity_n = 2 [10 (n + 2) (n + 1) a_n+ 2 + n (n - 1) a_n - n a_n + 20 a_2] x^n = 0 Implies 20 a_2 + 20 a_0 = 0, 60 a_3 - a_1 + 20 a_1 = 0, 10 (n + 1) (n + 2) a_n+ 2 = -(n (n - 1) - n + 20] a_n, Forall n greaterthanorequalto 2 Implies a_2