Unit3 - Subjective Questions

The general solution of a non-homogeneous linear differential equation with constant coefficients of the form consists of two parts:

1. Complementary Function (C.F.): This is the general solution of the corresponding homogeneous equation . It contains arbitrary constants equal to the order of the differential equation.
2. Particular Integral (P.I.): This is any specific solution of the non-homogeneous equation that contains no arbitrary constants.

Therefore, the complete general solution is given by:

When the given differential equation is , the Particular Integral (P.I.) is evaluated using the operator method as follows:

1. Write the P.I. as .
2. Substitute in , provided .
   Thus, .
3. Failure Case: If , then is a root of the auxiliary equation. Multiply the numerator by and differentiate the denominator with respect to .
   (provided ).
4. If , repeat the process: multiply by again (making it ) and differentiate the denominator again.

Step 1: Find the Complementary Function (C.F.)
The auxiliary equation is .
Solving this, , so the roots are .
.

Step 2: Find the Particular Integral (P.I.)
.
Substitute :
.

Step 3: General Solution
.

When or , the rule to find the Particular Integral is:

1. Write .
2. Replace with , provided .
   .
3. Failure Case: If , multiply the numerator by and differentiate the denominator with respect to . Then substitute again in the new denominator.
   Note: The substitution is specifically , not . Any remaining single terms in the denominator after substitution require rationalization to form terms.

When (a polynomial), the Particular Integral is found by expanding the inverse operator as a binomial series:

1. Write .
2. Take the lowest degree term common from to form an expression of the type .
3. Take this term to the numerator as .
4. Expand this expression using the binomial theorem up to the power , because the th and higher derivatives of will be zero.
   Common expansions used:
   
   
5. Operate this expansion on term by term.

The exponential shift theorem is used when the non-homogeneous term is the product of an exponential function and another function of , i.e., , where is any function of .

The rule states:

Steps:

1. Shift to the left of the operator.
2. Replace every in the operator with .
3. Operate the new operator on the remaining function using standard rules (e.g., if is a polynomial, sine, or cosine).

The Wronskian of two functions and , denoted as , is defined by the determinant:

Significance in Variation of Parameters:
In a second-order linear differential equation , let the complementary function be .
The Method of Variation of Parameters assumes the particular integral is , where and are functions of .
The solutions for and strictly depend on the Wronskian:


If , the functions and are linearly dependent, and the method cannot be applied. A non-zero Wronskian guarantees the independence of fundamental solutions and the existence of .

Consider the equation .
Let the C.F. be .
Assume P.I. is , where are functions of .
Differentiating :

We impose the condition: (Eq. 1)
Thus, .
Differentiating again:

Substitute into the original ODE:

Rearranging:

Since are solutions to the homogeneous equation, the terms in brackets are zero. We get:
(Eq. 2)
Solving Eq. 1 and Eq. 2 simultaneously for and using Cramer's rule:


Integrating gives and .

Step 1: Find C.F.
Auxiliary equation: .
.
Here, and . .

Step 2: Find Wronskian
.

Step 3: Find and

Step 4: Write P.I. and General Solution
.
General solution: .

Operator Method:

1. Applicability: Strictly applies to linear differential equations with constant coefficients.
2. Right-hand side : Works easily only for specific forms of like , , , , or products of these.
3. Complexity: Usually faster and simpler if is of a standard form.

Method of Variation of Parameters:

1. Applicability: Applies to linear differential equations with both constant and variable coefficients (provided the C.F. is known).
2. Right-hand side : Works for any continuous function (e.g., , , ), which operator methods cannot handle directly.
3. Complexity: Can be more tedious due to the need to evaluate integrals, which might be complex or non-elementary.

The Method of Undetermined Coefficients is a technique for finding the Particular Integral (P.I.) of a non-homogeneous linear differential equation with constant coefficients, .

Applicability:
It is applicable when the non-homogeneous term consists of functions whose derivatives form a finite set of linearly independent functions. This typically includes:

1. Exponential functions ()
2. Polynomials ()
3. Sines and cosines (, )
4. Finite sums and products of the above.

Methodology:
The method involves guessing the general form of the P.I. based on with unknown constant coefficients. This trial solution is substituted into the original differential equation, and the coefficients are determined by comparing like terms on both sides.

The rules for forming the trial solution depend on the form of :

1. Basic Rule:

   • If , guess .
   • If (polynomial of degree n), guess .
   • If or , guess .

2. Multiplication Rule: If is a product of the above types, the trial solution is the product of their respective guesses.

3. Modification Rule (Failure Case): If any term in the guessed trial solution is already a part of the Complementary Function (C.F.), the entire guessed trial solution must be multiplied by (or etc.) until no term in the new trial solution is a part of the C.F. The power of corresponds to the multiplicity of the root in the auxiliary equation.

Step 1: Check C.F. for overlap
Auxiliary equation: . Roots: .
. No overlap with polynomials.

Step 2: Form Trial Solution
Since is a quadratic polynomial, the trial solution is:

Derivatives:

Step 3: Substitute into ODE

Rearranging by powers of :

Step 4: Equate coefficients
:
:
:

Step 5: Write P.I.
.

Definition:
An Euler-Cauchy equation (or Cauchy-Euler equation) is a linear differential equation with variable coefficients of the form:

where are constants.

Transformation:
To convert it to an equation with constant coefficients, we use the substitution , which implies .
Let . The differential operators transform as follows:

Substituting these into the original equation yields a linear differential equation in terms of and with constant coefficients. This can then be solved using standard methods for C.F. and P.I.

Step 1: Transformation
Let , so and .
Substitute and .
The equation becomes:

Step 2: Solve the constant coefficient equation
Auxiliary equation: .
Factors: .
The solution in terms of is:

Step 3: Back substitute
Since and :
.

The defining characteristic of an Euler-Cauchy equation is that the power of matching the order of the derivative is attached as a coefficient (e.g., ).

The substitution (or ) is significant because it changes the independent variable such that the variable coefficients become constants.
By chain rule:

Taking higher derivatives continues to absorb the powers of into polynomial expressions of the new operator (like ).
This reduces a complex variable-coefficient problem into a much simpler constant-coefficient problem, allowing the use of auxiliary equations and standard Particular Integral techniques.

Step 1: Transformation
Put (). Let .
and .
The equation becomes:

Step 2: Complementary Function (C.F.)
Auxiliary equation: . Roots: .

Step 3: Particular Integral (P.I.)

Substitute :

Step 4: Complete Solution in

Substitute :
.

When evaluating or , we substitute . If the denominator becomes zero (i.e., ), the standard rule fails. This indicates that the non-homogeneous term is part of the Complementary Function.

Resolution:

1. Multiply the numerator by the independent variable .
2. Differentiate the denominator with respect to .
3. Apply the rule again: .
4. Depending on the new operator , multiply and divide by an appropriate factor to create terms in the denominator, and then substitute again.
5. If it fails again, repeat the process (multiply by and use the second derivative of ).

Legendre's Linear Differential Equation:
It is an equation of the form:

where are constants.

Generalization of Euler-Cauchy:
The Euler-Cauchy equation is a special case of Legendre's linear equation where and , reducing the terms to .

Solution Method:
Similar to Euler-Cauchy, it is reduced to constant coefficients using the substitution (so ). The derivatives transform as and , where .

Formula for :
If , where is a function of , the rule is:

Application:
Here , so , and .

First, operate on . Substitute :

Now, apply the bracket operator:


For the second term, substitute again:

Final P.I.:
.