Assuming a Power Series solution, y = ∑n=0^∞ Cn x^n, solve...

Assuming a Power Series solution, $y = \sum_{n=0}^{\infty} C_n x^n$, solve the differential equation \newline $y'' + y = 0$. Include the guaranteed radius of convergence and the recurrence relation in your \newline answer. \newline [20 marks]

Transcript

00:04 In this question we are given a power series first of all we are given a differential equation y dash is equal to 5 and we need to find out the power series solution of this by substituting y is equal to summation a is equal to 0 to infinity ck x to the power k that is our y is equal to c naught x 0 c 1 x to the power 1 c 2 x 2 and so on alright so y dash will be equal to here it will be k ck x to the power k minus alright summation k equal to 0 to infinity so this will be equal to 0 plus the derivative of c 1 x is c 1 derivative of c 2 x where is 2 c 2 x and so on alright now we need to substitute this into equation number 1 for substituting y dash we will be getting c 1 plus 2 c 2 x plus and so on is equal to c naught x to the power 0 plus let me write the third term here as well so that will be c 3 x 2 and here that will be 3 c 3 x square and so on so it will be 3 c 3 x and so on so here it means plus 3 c 3 x 3 and so on is equal to 5 into c naught plus c 1 x plus c 2 x square plus c 3 x cube and so on alright so now if we compare the coefficients let us first compare the constant coefficients so we will be getting c 1 equal to 5 c comparing the coefficients of x we will be getting 2 c 2 is equal to where the coefficient of x is 5 c 1 alright now comparing the coefficients of x square i forgot to write that term with x square here right coefficients of x square so here we will be getting 3 c 3 and here we will be getting 5 c 2 and so on so from here we have that c 1 equal to 5 c naught so from here we can get 5 into 5 c naught so c 2 is equal to 25 by 2 c naught alright similarly 3 c 3 is equal to 5 into value of c 2 is 25 by c naught so that implies c 3 is equal to 125 by 6 c naught and so on we can find out the value of all coefficient in terms of c so let us write y is that was equal to c naught plus c 1 x plus 2 c 2 x plus 3 c x square and so on so no it was c 3 no c 2 x square plus c 3 x cube and so alright so from here our values are c naught is c naught c 1 is 5 c naught x c 2 is 25 by 2 c naught c 3 is 125 by x 125 by 6 c naught x cube and so on so we can write y in term of c naught as 1 plus 5 x plus 25 by 2 x square…

Question

Assuming a Power Series solution, $y = \sum_{n=0}^{\infty} C_n x^n$, solve the differential equation \newline $y'' + y = 0$. Include the guaranteed radius of convergence and the recurrence relation in your \newline answer. \newline [20 marks]

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