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1. Let 2. Find 3. 4. Substitute the expressions of y, 5. Calculate 6. Put the values of |
Here we have- Let the solution of the given differential equation be- Since x = 0 is the ordinary point of the given equation- Put these values in the given differential equation- Equating the coefficients of various powers of x to zero, we get- Therefore, the solution is- |
Here we have- Let us suppose- Since x = 0 is the ordinary point of (1)- Then- And Put these values in equation (1)- We get-
Equating to zero the coefficients of the various powers of x, we get- And so on…. In general, we can write- Now putting n = 5, Put n = 6- Put n = 7, Put n = 8, Put n = 9, Put n = 10, Put the above values in equation (1), we get- |
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Case-1: when roots m1 and m2 are distinct and these are not differing by an integer- The complete solution in this case will be- Case-2: when roots m1 and m2 are equal- Case-3: when roots are distinct but differ by an integer- Case-4: Roots are distinct and differing by an integer, making some coefficient indeterminate- |
Here we have
Since x = 0 is a regular singular point, we assume the solution in the form So that
Substituting for y,
The coefficient of the lowest degree term = 0 in first summation only and equating it to zero. Then the indicial equation is Since The coefficient of next lowest degree term k = 1 in first summation and k = 0 in the second summation and equating it to zero. Equating to zero the coefficient of Or Which gives-
Hence for- Form m = 1/3- Hence for m = 1/3, the second solution will be- The complete solution will be-
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If x = 0 is a regular singularity of the equation.
Then the series solution is- Which is called Frobenius series. |
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If we put n = 0 then Bessel function becomes- Now if n = 1, then- |
As we know that- Now put n = 1/2 in equation (1), then we get- Hence proved. |
Put n = -1/2 in equation (1) of the above question, we get- |
As we know that- On differentiating with respect to x, we obtain- Putting r – 1 = s |
We have- Differentiating w.r.t. x, we get- |
We know that from formula first and second- Now adding these two, we get- Or |
We know that- On subtracting, we get- |
We know that- Multiply this by I.e. Or |
We know that- Multiply by Or |
We know that- The recurrence formula- On differentiating, we get- Now replace n by n -1 and n by n+1 in (1), we have- Put the values of |
We know that- from recurrence formula On integrating we get- On taking n = 2 in (1), we get- Again- Put the value of By equation (1), when n = 1 |
2. General solution of Bessel equation- |
Where
Where- For p non-integral, the general solution of Equation (2) is Thus the general solution of Equation (1) is When p is non-integral. |
The Legendre’s equations is- Now the solution of the given equation is the series of descending powers of x is- Here If n is a positive integer and The above solution is So that- Here Note- Legendre’s equations of second kind is
The general solution of Legendre’s equation is- Here A and B are arbitrary constants. |
Rodrigue’s formula can be defined as- |
We know that by Rodrigue formula- If n = 0, then it becomes- If n = 1, If n = 2, Now putting n =3, 4, 5……..n we get- ………………………………….. Where N = n/2 if n is even and N = 1/2 (n-1) if n is odd.
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By equating the coefficients of like powers of x, we get- Put these values in equation (1), we get-
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We know that- On integrating by parts, we get-
Now integrate (n – 2) times by parts, we get- |
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Formula-1: Fromula-2: Formula-3: Formula-4: Formula-5: Formula-6: |
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Now coefficient of Coefficient of Coefficient of And so on. Coefficient of The coefficients of Therefore-
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We know that
Equating the coefficients of |
And
Now multiply (1) by z and (2) by y and subtracting, we have-
Now integrate from -1 to +1, we get- |
On integrating by parts, we get- Now integrating m – 2 times, we get- |
2. The general solution of Legendre’s equation is- Here A and B are arbitrary constants. 3. Rodrigue’s formula can be defined as- 4. Orthogonality of Legendre polynomials- |