Linear Differential Equations

Definition

An equation containing a variable, its derivative and a few more functions of degree one is called a linear differential equation. The standard form of representing a linear differential equation is dy / dx + Py = Q. In this equation, x is an independent variable, and y is the dependent variable. We find derivatives of dependent variables only with respect to an independent variable.

Every linear differential equation can be solved using a particular formula for its solution. Some examples of linear differential equations are:
1. dy / dx + y = sin x
2. dy / dx + (-27y) / x = x2.e-x

General solution formula of a linear differential equation

The general solution of the differential equation dy / x + Py = Q is given by:

y . (I.F) = ∫(Q . (I.F) . dx) + Cy . (I.F) = ∫(Q . (I.F) . dx) + C

Here, I.F. is the integrating factor and is given by: e ∫P . dx e ∫P . dx.

We can interchange x and y if y is our independent variable and x is a dependent variable depending upon the question.

Derivation of the solution of a differential equation

We know, the first order differential equation is represented as: dy /dx + Px = Q

We need to multiply both sides by a function in x, say g (x), such that the right hand side of the above equation gets a derivative in y. g (x). Therefore,
d /dx (y . g (x)) = y . g (x)
g (x) . dy / dx + P . g (x) . y = Q . g (x)
g (x) . dy / dx + P . g (x) y = d / dx (y . g (x)]
g (x) . dy / dx + P . g (x) . y = g (x) . dy / dx + y . g' (x)
P . g (x) = g' (x)
P = g' (x) / g(x)

Integrating both sides with respect to x, we get,
∫P . dx = ∫g′ (x) / g (x) . dx

∫P . dx = log (g(x))

g (x) = e∫P . dx

The function g (x) = e∫P . dx is the Integrating Factor (I.F) of the given linear differential equation. Substituting the value of g (x) in linear differential equation, we get,

e∫P . dx . dy / dx + P e∫P.dx y = Q . e∫P . dx

d / dx (y . e∫P . dx) = Q e∫P . dx

Integrating both sides, with respect to x, we get,

y . e∫P . dx = ∫(Q . e∫P . dx . dx

y = e−∫P . dx .∫(Q . e∫P.dx . dx) + C

The above expression is the general solution of the linear differential equation.

Steps to solve a linear differential equation

1. Simplify the entire differential equation and write it down in the form dy / dx + Py = Q.
2. Find the integrating factor of the given differential equation.
3. Write down the solution of the linear differential equation as: y (I.F) = ∫(Q × I.F) . dx + C.

Example: Find the general solution of the differential equation x dy - (y + 2x2) . dx = 0
Solution:

We can write the simplified form of the differential equation as:
dy / dx - y/x = 2x

Comparing this with the differential equation dy/dx + Py = Q we have the values of P = -1/x and Q = 2x.

Hence, the integration factor IF = e ∫(−1/x) . dx

= e−log x =1/x
We need to solve the equation as:

y 1/x = ∫2x . 1/y . dx + c
y/x = ∫2 . dx + c
y/x = 2x + c
y = 2x2 + xc, which is the required solution of the differential equation.