Unit-2: 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.