# Linear Differential Equations

**Linear Differential Equation of nth Order**

*Linear differential equation* is in the form of Ly=f, where ‘L’ is a linear operator, ‘Y’ is a unknown function and ‘F’ is a known function of a same nature. In mathematical point of view first-order linear differential equation are those equation that can be kept in form.

1. **A differential equation of the form**

where are constant and Q in any function of x or constant is called linear differential equating of nth order with constant co-effieient.

**Note**:

(1)

(2) are equation of second order.

2.We use

3.**Solution f the equation f (D) y = Q**

**Case 1**:

If are independent solutions of the equation , or f (D)y = 0 then, will also be solution of the equation.

**Case 2**:

If be a general solution of the equation f (D)Y = 0 and be particular solution of f (D)Y = Q then is the general solution of f (D)y = Q.

4.**If m replace D by m then the equation**

reduces to but

So, (i) is called auxiliary equation. Equation (i) is quadratic in m and must have two roots.

**Case 1**: *auxiliary equation having different and real roots, equal roots, imaginary roots.*

Roots | Solution | General solution |

5. **Some useful relations to find P.I.**

where m is positive integer

In this case we have to use binomial theorem and then find derivatives successively, for this we need following relations.

, v is function of x or constant.

v is function of x.

[=real part of is [co-efficient of I in

10. **Homogenous Linear equation**

An equation of the form where are constant and Q is function of x, is called homogenous linear equation

**Note:**

is homogeneous linear equation of second order.

11. **Equation Reducible to the Homogeneous Linear Form**.

The equation of the form

is reduced to the homogeneous linear form by putting;

A + bx = t so that the equation can be directly reduced to one with constant co-efficient by putting

Related posts:

- Linear Differential Equation Formula Linear Differential Equation Formula Some of the important formulas...
- Bernoulli’s Equation The Bernoulli’s differential equation is written in the form :...
- Application of Differential Equation Application of Differential Equation Differential equation can be defined...
- Exact differential equation Exact differential equation A differential equation is a equation used...
- Constant acceleration & Constant acceleration equations Constant acceleration. And Constant Constant acceleration equation. Derivation of constant...