Unit-3: Linear Differential Equations

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1. Introduction to Linear Differential Equations

A linear differential equation is an equation that involves the derivatives of a function and can be expressed in the form:

a_n(x)y⁽ⁿ⁾ + a_n-1(x)y^(n-1) + ... + a₁(x)y' + a₀(x)y = f(x)

Here, f(x) is the non-homogeneous part (if present), and a₀, a₁, ..., a_n are functions of x or constants.

2. Solution of Linear Differential Equations

The general solution of a linear differential equation consists of two parts:

• Complementary Function (C.F.): The solution of the corresponding homogeneous differential equation (f(x) = 0).
• Particular Integral (P.I.): A specific solution of the non-homogeneous differential equation.

Hence, the general solution is given by:

y = C.F. + P.I.

3. Linear Dependence and Independence of Solutions

A set of functions {y₁, y₂, ..., y_n} is said to be linearly independent if no linear combination of these functions equals zero, except for the trivial case when all constants are zero:

c₁y₁ + c₂y₂ + ... + c_ny_n = 0 → c₁ = c₂ = ... = c_n = 0

If any non-zero combination exists, the functions are linearly dependent.

4. Method of Solution Using Differential Operator

The differential operator D is defined as:

D = d/dx

This allows us to write differential equations in a simplified form:

F(D)y = g(x)

Where F(D) is a polynomial in D. This makes solving differential equations more systematic.

5. Solution of Second Order Homogeneous Linear Differential Equations with Constant Coefficients

A second-order homogeneous linear differential equation with constant coefficients has the form:

ay'' + by' + cy = 0

Its characteristic equation is:

ar² + br + c = 0

The nature of the roots of this characteristic equation determines the solution:

• Distinct Real Roots: y = c₁e^r₁x + c₂e^r₂x
• Repeated Real Roots: y = (c₁ + c₂x)e^rx
• Complex Roots: y = e^αx(c₁cos(βx) + c₂sin(βx))

6. Solution of Higher Order Homogeneous Linear Differential Equations with Constant Coefficients

For a higher-order homogeneous linear differential equation of the form:

a_ny⁽ⁿ⁾ + a_n-1y^(n-1) + ... + a₁y' + a₀y = 0

We write the characteristic polynomial:

a_nrⁿ + a_n-1r^n-1 + ... + a₁r + a₀ = 0

The general solution is formed using the corresponding characteristic roots, applying similar principles as for second-order equations.

7. Solution of Non-Homogeneous Linear Differential Equations

7.1 Using Operator Method

We write the equation as F(D)y = g(x). First, solve the homogeneous part F(D)y = 0 to get the complementary function. Then find the particular integral using the inverse operator method: P.I. = F(D)^-1g(x).

7.2 Method of Undetermined Coefficients

Assume a particular form of the solution based on the form of g(x). Substitute into the differential equation to determine the unknown coefficients.

7.3 Method of Variation of Parameters

For the equation y'' + p(x)y' + q(x)y = g(x), the particular integral is given by:

y_p = u₁(x)y₁ + u₂(x)y₂

Where u₁ and u₂ are functions determined using specific formulas.

8. Solution of Euler-Cauchy Equation

An Euler-Cauchy equation is of the form:

x²y'' + axy' + by = 0

We substitute y = x^r and form the auxiliary equation to determine the solution.

9. Solution of Simultaneous Differential Equations Using Operator Method

For a system of simultaneous linear differential equations, we use operator notation and solve using elimination or matrix methods.

Example: Dx + y = e^x, Dy - x = e^x