Chapter 5, Section 5.2, Question 2 In the Problem: • a. Seek power series solutions of...
1. (20 pts.) In the following Problems: (a) Seek power series solutions of the given differential equation about the given point xo ; find the recurrence relation. (b) Find the first four terms in each of two solutions yi and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(xo), show that yı and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution. i) y" +k+x+y = 0, 40...
In each of Problems 3 and 4 (a) Seek the power series solutions of the given differential equation about the given point ro: find the recurrence relation. (b) Find the first four terms in each of two solutions vi and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W (, 2)(o), sow that n and y2 form a fundamental set of solutions (d) If possible, find the general term in each solution. 3. Exercise 5.2 #5. 4....
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)...
Use a power series centered about the ordinary point x0 = 0 to solve the differential equation (x − 4)y′′ − y′ + 12xy = 0 Find the recurrence relation and at least the first four nonzero terms of each of the two linearly inde- pendent solutions (unless the series terminates sooner). What is the guaranteed radius of convergence?
1-find the recurrence relation using power series solutions. 2-find the first four terms in each of two solutions y1 and y2 3-by evaluating wronskian w(y1,y2) show that they from a fundamental solution set. Iy yry 0, zo = 1
Seek power series solution of the given differential equation about the given point x0; find the recurrence relation.(1-x)y'' + y = 0; x0 = 0
differential equations 1 +.. 8 Find two power series solutions of the given differential equation about the ordinary point x = 0. (x2 + 1)" - 6y = 0 O Y1 = 1 + x2 + 3x4 xo and Y2 = x = x + 3x3 16 O x1 = 1 + 3x2 + x4 – xo + and y2 = x + x3 O Y1 = 1 + 3x2 + 5x* + 7x® + ... and y2 = x...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
(1 point) It can be shown that yı = e-4x and y2 = xe-4x are solutions to the differential equation y + 8y +16y=0 on the interval (-00, 00). Find the Wronskian of yn y (Note the order matters) W(y1, y2) = Do the functions yn y form a fundamental set on (-00,00)? Answer should be yes or no
10.5.3 Consider the defining differential equation for the Hermite polynomials do and solve it by the series solution method for functions Hn(x such that Hx)exp(-x2/2) can be normalized In your solution (i) find a recurrence relation between the coefficients of the power series solutions [Note: this (ii) show that Hn(x)exp(x/2) wll not be normalizable unless the power series terminates (ii) choosing co 0 or 1 and c0 or 1, find the first 5 power series solutions of the equation. relation...