Unit 2: Linear differential equation-I - Subjective Questions

Definition: A differential equation is called linear if the dependent variable and all its derivatives occur only to the first degree and are not multiplied together or involved in nonlinear functions.

The general form of an th-order linear differential equation is

where the coefficients and the function depend only on .

Linear example:

Nonlinear examples:

• contains a squared derivative.
• contains a product of and its derivative.
• contains a nonlinear function of .

Thus, linearity requires that and its derivatives appear linearly, with coefficients independent of .

A first-order linear differential equation is written as

Step 1: Determine the integrating factor.

The integrating factor is

Step 2: Multiply the equation by the integrating factor.

After multiplication,

The left-hand side is the derivative of . Therefore,

Step 3: Integrate both sides.

Hence, the general solution is

The equation is already in the standard form

where and .

Integrating factor:

Multiplying the equation by gives

The left-hand side is

Integrating,

Therefore,

This is the general solution of the given linear differential equation.

Consider the homogeneous linear differential equation

where is a linear differential operator.

If are solutions of the equation, then any linear combination

is also a solution, where are arbitrary constants.

Proof: Since is linear,

Because each is a solution, . Hence,

This property is called the principle of superposition. It is valid only for linear homogeneous equations and is used to construct the general solution from linearly independent solutions.

Let be solutions of a homogeneous linear differential equation.

They are linearly dependent on an interval if constants , not all zero, exist such that

for every point in the interval.

They are linearly independent if the relation

implies

For an th-order homogeneous linear differential equation, the general solution requires linearly independent solutions:

If the solutions are dependent, some solutions are redundant and cannot provide the required number of arbitrary constants. Therefore, linear independence is essential for obtaining the complete general solution.

For two functions and , the Wronskian is defined by

Test:

• If at some point in an interval, then and are linearly independent on that interval.
• If throughout the interval, the functions may be linearly dependent; for solutions of a suitable linear differential equation, this indicates dependence under the usual existence conditions.

Example: Let

Then

Since , the functions are linearly independent. Thus, they can be used as two independent components of the general solution of a second-order equation.

The differential operator is defined by

Consequently,

and, in general,

A constant-coefficient linear differential equation such as

can be written as

where

For example,

becomes

This notation simplifies the solution of higher-order equations.

Consider the nonhomogeneous equation

Its complete solution is

Complementary function: The complementary function is the general solution of the associated homogeneous equation

It is obtained from the auxiliary or characteristic equation

Particular integral: The particular integral is any one solution of

In operator notation, it is represented by

where the inverse operator is evaluated by suitable rules.

Therefore,

The complementary function contains arbitrary constants, while the particular integral does not contain arbitrary constants.

Consider

where are constants and .

Assume a trial solution

Then

Substitution gives

Since ,

This is the auxiliary equation. Let its roots be and .

Case 1: Distinct real roots

If , then

Case 2: Repeated root

If , then

Case 3: Complex roots

If , then

The differential equation is

Assume . The auxiliary equation is

Factoring,

Hence,

The roots are distinct and real. Therefore, the complementary function is

Thus, the general solution is

The two exponential terms are linearly independent because their exponents are different.

The equation is

Its auxiliary equation is

which factors as

Thus, is a repeated root. For a repeated root of multiplicity two, the independent solutions are

Therefore, the general solution is

The factor is necessary to produce a second linearly independent solution.

The auxiliary equation is

Using the quadratic formula,

Thus, the roots are of the form

where and .

For complex conjugate roots, the real-valued general solution is

Hence,

The exponential factor controls the decay, while the sine and cosine terms represent oscillation.

For a constant-coefficient equation , suppose the auxiliary equation has a root repeated times. The corresponding linearly independent solutions are

Therefore, the contribution to the complementary function is

For example, if is repeated three times, the corresponding part is

The powers of ensure linear independence and provide the required number of arbitrary constants. If several distinct roots have different multiplicities, the corresponding contributions are added together.

The operator equation is

The auxiliary equation is

Its roots are

The root is repeated twice. Therefore, its solutions are

The root gives the solution . Hence, the complementary function is

There are three arbitrary constants, as expected for a third-order differential equation.

Suppose the auxiliary equation has complex conjugate roots

The corresponding complex solutions are

Using Euler's formula,

we obtain

The real and imaginary parts are both real solutions:

Therefore, the general real solution is

These two solutions are linearly independent when .

The auxiliary equation is

Grouping terms,

so

The roots are

The real root gives . The complex roots give

Therefore, the general solution is

The three terms are linearly independent and correspond to the third-order equation.

Consider the equation

The solution procedure is:

• Write the equation in operator form:

• Form the auxiliary equation by replacing with :

• Find all roots of the auxiliary equation, including their multiplicities.

• Construct the complementary function:

  • For distinct real roots , use .
  • For a repeated root of multiplicity , use .
  • For complex roots , use .

• Combine all independent parts to obtain the general solution.

The number of arbitrary constants must equal the order of the differential equation.

The form of the complementary function depends on the roots of the auxiliary equation.

For higher multiplicity, a repeated real root of multiplicity contributes

Distinct real roots produce exponential growth or decay. Repeated roots produce exponential terms multiplied by powers of . Complex roots produce oscillatory sine and cosine terms multiplied by an exponential factor.

The Wronskian is

Here,

and

Therefore,

Since

for every real , the Wronskian is nonzero. Hence, and are linearly independent.

An th-order differential equation requires integrations to obtain its general solution. Each integration introduces one arbitrary constant. Hence, the general solution normally contains arbitrary constants.

For a constant-coefficient homogeneous equation,

where has degree , the auxiliary equation has roots when counted with multiplicity. Each root contributes independent solutions whose total number is .

For example:

• Three distinct roots contribute three exponential solutions.
• A repeated root of multiplicity three contributes , , and .
• A complex conjugate pair contributes two real solutions involving sine and cosine.

Thus, the complementary function has the form

where are linearly independent solutions. The constants are determined by independent initial or boundary conditions.