#415 Consider the differential equation xy + y - xy = 0

Consider the differential equation xy + y - xy = 0 - Math

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Free Chegg Question

Consider the differential equation,

The objective is to find the basis of solution by the Frobenius method.

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Free Chegg Answer

1. Step 1 of 10

   Consider the differential equation,

   

   The objective is to find the basis of solution by the Frobenius method.

2. Step 2 of 10

   Let . Then,

   ,

   Substitute the above values in the equation, get,

   

   Separate the terms for .

   

   Replace  in the first and second series and  in third series.

   

3. Step 3 of 10

   Continuation to the above step,

   

   

   

   Compare the coefficients on both sides to get,

    

   And,

   

4. Step 4 of 10

   Since  yields nothing, take

     

   Hence the indicial roots are, .

   For  the recurrence relation becomes,

   

   And,

   

   For ,

   

5. Step 5 of 10

   For ,

   

   For ,

   

    

6. Step 6 of 10

   For ,

    

   

   For ,

   

   And so on.

7. Step 7 of 10

   Therefore, the first solution is,

   

   

   That is, the solution is,

   

8. Step 8 of 10

   Use reduction of order method to obtain the second solution.

   Formula to obtain second solution of standard form,

    ,  as first second solution

   is, 

9. Step 9 of 10

   write the given differential equation in standard form.

    [Original equation]

    standard form

   Comparing it with get

   Hence the second solution of this equation is,

   

   

10. Step 10 of 10

    Continuation to the above step,

    

    

    

    Hence the second solution is,