Unit – 3
Series solution of differential equation
- Solve in series the equation
Ans.
Since x=0 is the ordinary point of the equation (1)
Then
Substituting in (1) we get
Equating to zero the coefficient of the various powers of x we obtain
Substituting these values in (2) we get
2. Solve.
Ans. Let,
Substituting the value of in the given equation we get
Where the first summation extends over all values of K from 2 to
And the second from K =
Now equating the coefficient of equal to zero we have
For K =4
3. Solve.
Ans. Let
Substituting for in the given differential equation
Equating the coefficients of various powers of x to zero we get
4. Legendre’s equation is
And
Prove that
Ans. We know that
Put n=2
5. Prove that .
Ans. We know
+
Put x = 1 both sides we get
Equating the coefficient of on both sides we get
6. Prove that
Ans. We know
Differentiating with respect to z we get
Multiplying both sides by we get
Equating the coefficient of from both sides we get
7. Solve. Statement
Proof. Let is a solution of
is the solution of
Multiplying (1) by z and (2) by y and subtracting we get
Now integrative -1 to 1 we get
Now we have to prove that
We know that,
Squaring both sides we get
Integrating both sides between -1 to +1 we get
on both sides we get
here n = m
8. Prove that
Ans. The Recurrence formula is
Pn+1+nPn-1
Replacing n by (n+1) and (n-1) we have
Multiplying (1) and (2) and integrating in the limits -1 to 1 we get
(By orthogonality property)