The differential equation y = = ry, with initial conditions y(0) = 1,y'(0) = 0 looks...

 

Transcribed Image Text:

The differential equation y" = = ry, with initial conditions y(0) = 1,y'(0) = 0 looks harmless. However, it does not have a solution that can be written as an elementary function. Let's try to solve this equation in terms of a power series y(x) = ao + a1x + a2x² + azx + agx? +.... a) Find the first 10 coefficients of the power series by writing both sides of the differential equation in terms of the power series and comparing coefficients (coefficients of corresponding powers of a have to be equal!). [10 marks] b) Describe the pattern underlying the coefficients. What is the general term a,? [10 marks] c) Apply the ratio test to two consecutive non-zero terms of the power series to calculate its radius of convergence. [10 marks] d) Plot the first four different partial sums of the power series in the interval [-3.5, 3.5].1 [10 marks] The differential equation y" = = ry, with initial conditions y(0) = 1,y'(0) = 0 looks harmless. However, it does not have a solution that can be written as an elementary function. Let's try to solve this equation in terms of a power series y(x) = ao + a1x + a2x² + azx + agx? +.... a) Find the first 10 coefficients of the power series by writing both sides of the differential equation in terms of the power series and comparing coefficients (coefficients of corresponding powers of a have to be equal!). [10 marks] b) Describe the pattern underlying the coefficients. What is the general term a,? [10 marks] c) Apply the ratio test to two consecutive non-zero terms of the power series to calculate its radius of convergence. [10 marks] d) Plot the first four different partial sums of the power series in the interval [-3.5, 3.5].1 [10 marks]

Answer rating: 100% (QA)

09 28 WK 190th DAY FRIDAY JULY 10 Let be solution to 0 Yo 10 5 nn1 ... View the full answer