Weshal Dow sake up the case when the roots of indicial equation are equal. Case

Question

Weshal Dow sake up the case when the roots of indicial equation are equal. Case 3: Indicial roots are equal Inthis case, the iollowing rule works: Rule: If indicial equation has two equal roots k, k obtain twolinearly independent solution by sbstituting this value of kin y and We illustrate this rule through an example. Example 10: Solve, in series the differential equation Let the series solution of Eqn (88) be of the form Putting these values of y,y.y" in Eqnuss, we get which is an identity. Equating to zero the coefficient of least power x.inidentity c91) Thus indicial equation has equal roots. Again from Eqn (91 the recurrence relation is obtainod as Again equating the coefficient of x' in Eqnol) to zero, we get ci(k+1)3 0, but since k -0,k +1.0 and hence ci -0. For e, 0, we get from Eqnu(93), ci ci -0 cs c, For m 2,4,6.... Eqn(93) yields (k 2) (k 4) Putting these values of c's in Egn (89), we get Putting k w 0 in Eqn (95), we get simplifying, we get

Explanation / Answer

you are right ...as both differential equations are different 88 and 97 are both different equations ....that's why you are struck with the answer...if you are considering equation 88 just simply differentiate equation 95 and substitute k=0 you will get answer.

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