First Order Linear Equation

A first-order differential equation is an ordinary differential equation in which any derivatives with respect to the independent variable have an order less than 1. If the variables m and n can be separated then a first-order equation is said to be separable. By separating the variables and then integrating both sides of the equation with respect to m, one can solve such equations.

Linear Differential equation of first order

A differential equation of type

n’ + a(m)n = f(m),

Where,

an (m) and f(m) are continuous functions of m, which is called a linear equation of first order. Linear equation of first order can be solved by using two methods:

• Method of variation of a constant; and
• Using an integrating factor

Method of variation of a constant

At first, one need to find the general solution of the homogeneous equation:

n’ + a (m)n = 0

The general solution of the homogeneous equation contains a constant of integration C. The constant integer C needs to be replaced with a certain function C (m). By substituting this solution into the non-homogeneous differential equation, one can determine the function C (m).

Using an Integrating Factor

When a differential linear equation is portrayed in a standard form:

n’ + a (m)n = f(m),

Then the factor integrated in defined by the given formula

u (m) = exp (⌠a (m)dm

By multiplying the equations of the left-hand side of the equation by the integrating factor u (m) converts the left side into the derivative of the product n (m) u (m).

The differential equation’s general solution is expressed as follows:

n = ⌠u (m) f(m) dm + C / u (m)

C is the arbitrary constant

Nonlinear differential equations of the first order

Although there is no general method for the solution is available for first-order nonlinear equations, there are several cases of physically relevant nonlinear equations which can be solved analytically:

• Separable equations

dn / dm = f(m) / g(n)

• ⌠g(n) dn = ⌠f(m)dm

For example:

dn / dm = n²e^x

• ⌠dn / n² = ⌠e^mdm
• -1 / n = e^x + c

Or,

n = -1 / (e^{x + c})

• Almost separable equations

dn /dm = f(am + bn)

Change variables:

z = am + bn

• dz / dm = a + b(dn / dm)
• dz / dm = a + bf(z)
• x = ⌠1 / {a +bf(z)} dz

For example:

Dn / dm = (-4m + n) ²

Z = n – 4m

• dz / dm = -4 + dn / dm = z² – 4

m = ¼ ln(z-2 / z+2) + C

• n = 4m + 2 (1+ke^4x) / (1- ke^4x)      [k is constant here]

• Homogeneous equations

dn / dm = f (n / m)

The equation is invariant under m→ sm, n→ sn

Solution

n = vm

•  n’ = v’m + v

i.e. v’ = 1 / m {f(v) –v}

• ⌠dv / f(v) –v = ⌠dm /m = ln m + constant

For example:

mn (dn/dm) – n² = (m + n)²e^-n/m

Change variables

n = vm

• n’ = v’m + v.

(v’m + v) – v =  {(1 + v)² / v } e^-v

• ln m = ⌠e^vvdv / (1+v)²

To evaluate integral change variables

u = 1+ v

e^-1  ⌠(1/u – 1/u²)e^u du = e^-1[e^u/u]

i.e  ln m = e^n/m / (1+ n/m)

• Homogeneous but for constants

dn / dm = m+ 2n + 1 / x + y+ 2

m = m’ + a,  n = n’ +b

• dn/dm = dn’ / dm + dn’ / dm’ = dm’ / dm = dn’ / dn’
• dn’ / dm’ = m’ + 2n’ + 1 + a +2b / m’ + n’ + 2 + a +b

[1+a+2b = 0 and 2+a+b=0]

[a = -3 and b = 1]

• dn’ / dm’ = m’ + 2n’ / m’ + n’

• The Bernoulli Equation

dn / dm + P(m)n = Q(m)yⁿ,    n≠1

T solve, change variable to

z = n^1-n

• dz / dm = (1 – n)n^-n(dn / dm)

Gives the equation [dz /dm + (1-n)P(m)z = (1 – n)Q(m)] {First Order Linear}

For example:

n’+n= y^2/3

z = y^1-n = n^1/3

• z’ + z /3 = 1/3   {First order Linear}

Integrating factor e^m/3

• ze^m/3 ⌠e^m/3 dm / 3

∴ z = n^1/3 = 1 + ce^-x/3

Conclusion

This is to conclude that a first-order differential equation is an ordinary differential equation in which any derivatives concerning the independent variable have the order of less than 1. If the variables m and n can be separated then a first-order equation is said to be separable. By separating the variables and then integrating both sides of the equation with respect to m, one can solve such equations.