Unit – 4
Differential equations
The general form of linear differential equation of second order is
Where p and q are constants and R is a function of x or constant.
Differential Operator
D stands for operation of differential i.e.
stands for the operator of integration.
stands for operation of integration twice.
Thus, 
Note:- Complete solution = complementary function + Particular integral
i.e. y=CF + PI
Method for finding the CF
Step1:- In finding the CF right hand side of the given equation is replaced by zero.
Step 2:- Let 
be the CF of
Putting the value of 
in equation (1) we get
It is called auxiliary equation.
Step 3:- Roots Real and Different
If 
are the roots the CF is 
If 
are the roots then 
Step 4- Roots Real and Equal
If both the roots are 
then CF is
If roots are 
Example: Solve 
Ans. Given, 
Here Auxiliary equation is
Solve: 
Or,
Ans. Auxiliary equation are 
Note: If roots are in complex form i.e. 
Solve: 
Ans. Auxiliary equation are 
Solve. 
Ans. Its auxiliary equation is
Solution is 
Solve. 
Ans. The auxiliary equation is
Hence the solution is
Rules to find Particular Integral
Case 1: 
If, 
If, 
Solve: 
Ans. Given, 
Auxiliary equation is 
Case2: 
Expand 
by the binomial theorem in ascending powers of D as far as the result of operation on 
is zero.
Solve. 
Given, 
For CF,
Auxiliary equation are 
For PI
Case 3: 
Or,
Solve: 
Ans. Auxiliary equation are 
Case 4: 
Solve. 
Ans. AE= 
Complete solution is 
Solve 
Ans. The AE is 
Complete solution y= CF + PI
Solve. 
Ans. The AE is 
Complete solution = CF + PI
Solve. 
Ans. The AE is 
Complete solutio0n is y= CF + PI
Find the PI of 
Ans. 
Solve 
Ans. Given equation in symbolic form is
Its Auxiliary equation is 
Complete solution is y= CF + PI
Solve. 
Ans. The AE is 
We know, 
Complete solution is y= CF + PI
Solve. Find the PI of (D2-4D+3)y=ex cos2x
Ans. 
Solve. (D3-7D-6) y=e2x (1+x)
Ans. The auxiliary equation i9s
Hence complete solution is y= CF + PI
Working Rule
Step 1 Let 
be the solution of the given differential equation.
Step 2: Find 
etc.
Step 3: Substitute the expression of 
in the given differential equation.
Step 4: Calculate 
coefficient of various powers of x by equating coefficients to zero.
Step 5: Substitute the values of 
in the differential equation to get the required series solution.
Example. 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
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
Solve. 
Ans. Let 
Substituting for 
in the given differential equation
Equating the coefficients of various powers of x to zero we get
Legendre’s equation is 
And 
Prove that 
Ans. We know that 
Put n=2
Prove that 
.
Ans. We know
+
Put x = 1 both sides we get
Equating the coefficient of 
on both sides we get
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
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
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)
The Bessel equation is
Bessel function of first kind
Bessel function of second kind
Recurrence Formula
1) xJn'=nJn-xJn+1
2) 
3) 
4) 
5) 
6) 
Prove that (1) 
Ans. We know
(b) Prove that 
Ans. We know that
(3) Prove that 
Ans. We know that
Jn(x)=
If n = 0
If n = 1
Note General solution of Bessel Equation
References
1. Ordinary and Partial Differential equations by J. Sihna Ray and S Padhy, Kalyani Publishers
2. Advance Engineering Mathematics by P.V.O’NEIL, CENGAGE
3. Ordinary Differential Equation by P C Biswal , PHI secondedition.
4. Engineering Mathematics by P. S. Das & C. Vijayakumari, Pearson. N.B:Thecourseisof3creditwith4contacthours.
