#416 Consider the differential equation as follows

Consider the differential equation as follows - Math

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Consider the differential equation as follows:

 …… (1)

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Free Chegg Answer

1. 1. Step 1 of 4

      Consider the differential equation as follows:

       …… (1)

      The objective is to determine the basis of solutions of equation (1) by Frobenius Method.

      Assume that the solution of equation (1) is,

      

      Differentiate y w.r.t. x.

      

      Differentiate  w.r.t. x.

      

   2. Step 2 of 4

      Substitute the expressions of  in equation (1).

      

      

       …… (2)

      The smallest power of x is  occurring in second term when .

      Thus, equate the coefficients of  on both sides to obtain the indicial equation,

      .

      The indicial equation  has roots .

      For distinct roots not differing by an integer, by Frobenius method, a basis is as follows:

      .

   3. Step 3 of 4

      First solution:

      When .

      To determine the coefficients of equation 

      Substitute  in equation (2) and simplify as follows:

      

      

      Equate the coefficients of powers of x on both sides.

      .

      This implies .

      Assume that .

      The expression  implies that 

      Thus, the basis is,

      

       

   4. Step 4 of 4

      Second solution:

       

      When .

      To determine the coefficients of equation 

      Substitute  in equation (2) and simplify as follows:

      

      

       

      Equate the coefficients of powers of x on both sides.

      .

      This implies .

      Assume that .

      The expression  implies that .

      Thus, .

      Thus, the basis is,

      

      Hence, the required solution of the differential equation is .