(1 point) Frobenius' method: finding solutions as generalized power series Example: Consider the equation Tºg +...
Find two power series solutions of the given differential equation about the ordinary point x = 0. y′′ − 4xy′ + y = 0 Find two power series solutions of the given differential equation about the ordinary point x = 0. y!' - 4xy' + y = 0 Step 1 We are asked to find two power series solutions to the following homogenous linear second-order differential equation. y" - 4xy' + y = 0 By Theorem 6.2.1, we know two...
In this exercise we consider the second order linear equation y" therefore has a power series solution in the form 4y = 0. This equation has an ordinary point at x = 0 and We learned how to easily solve problems like this in several different ways but here we want to consider the power series method (1) Insert the formal power series into the differential equation and derive the recurrence relation Cn-2 for n - 2, 3, NOTE co...
(1 point) In this exercise we consider the second order linear equation y" + series solution in the form y = 0. This equation has an ordinary point at x = 0 and therefore has a power y = cmx". n=0 We learned how to easily solve problems like this in several different ways but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation Cn...
2. Find power series solutions y z" Σ anr" of the following equation centered at 0 where-0 is a regular singular point. (a) Find the indicial equation for r, and solve for the two roots. Note that the indicial equation can be obtained from the coefficients of the term Pick the larger root and find the first seven terms of your power series solutions, i.e., (b)
Need some help with SERIES SOLUTION - 2nd ORDER EQUATION For the differential equation, (1) a. Calculate the indicial equation for the power series solution (Answer in a quadratic polynomial in terms of c.) b. Calculate the solutions of the indicial equation found above. c. Calculate the point from the above equation (1) as i. ORDINARY POINT ii. REGULAR SINGULAR POINT iii. IRREGULAR SINGULAR POINT We were unable to transcribe this imagey-Σ@m(z _ 4)nte We were unable to transcribe this...
1 point) Consider the differential equation which has a regular singular point at x = O. The indicial equation for x 0 is r1/2 r+ 0 =0 and r O with roots (in increasing order) r1/2 Find the indicated terms of the following series solutions of the differential equation: (a) y = x, (94 (b)y-x(5+ The closed form of solution (a) is y = xtO r3+ 1 point) Consider the differential equation which has a regular singular point at x...
(6 points) Use the method of Frobenius to obtain linearly independent series solutions about x = 0. 3xy" – 1.54' + 2y = 0. Use an initial index of k = 1 to develop the recurrence relations. The indicial roots are(in ascending order) rı = 4.5/3 ,12 = Corresponding to the smaller indicial root, the recurrence relation of the solution is given by C = Xck-1. The initial index is k = The solution is yı = c (az xb1...
(1 point) Consider the differential equation 2x(x )y"3 - 1)y -y0 which has a regular singular point atx 0. The indicial equation for x 0 is 2+ 0 r+ with roots (in increasing order) r and r2 Find the indicated terms of the following series solutions of the differential equation: x4. (a) y =x (9+ x+ (b) y x(7+ The closed form of solution (a) is y (1 point) Consider the differential equation 2x(x )y"3 - 1)y -y0 which has...
1) Find two power series solutions of the differential equation (x² + 1)y" – xy' + y = 0 about the ordinary point x = 0. Hint: Check Examples 5 and 6 in 6.2 Example 6 Power Series Solution Solve (x + 1)," + xy - y = 0. Solution As we have already seen the given differential equation has singular points at = = ti, and so a power series solution centered at o will converge at least for...
Using the method of Frobenius obtain two linearly independent solutions to the differential equation (two power series solutions first four terms) 2x2y'' + (x-x2)y' - y = 0