Linear First-Order Differential Equation

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general formulation

example

graph

new equation

c

1

-1

A linear differential equation of the first order has the form

y'+ P y = Q.

First we solve the corresponding homogeneous equation

y' + P y = 0, dy/dx + P y = 0, dy/y = -P dx, log(y) = log(C) - ∫ P dx.

The general solution of the equation is

y

h

= C

-∫Pdx

e

,

y

h

= C v(x), v(x) =

-∫Pdx

e

.

The method called "variation of parameters" consists of finding a particular solution of the original equation in the form

y

p

= C(x) v(x).

Substituting

y

p

we get:

C'(x) v(x) + C(x) (v'(x) + P v(x)) = Q.

Since v(x) is a solution of the homogeneous equation

C'(x) v(x) = Q, C'(x) = Q / v(x), C(x) = ∫ Q / v(x) dx.

The general solution of the original equation is the sum of the general solution of the homogeneous equation and of the particular solution:

y =

y

h

+

y

p

.

The method is valid even in case of an equation of the form

R y' + P y = Q.

The Demonstration explains the "variation of parameters" method of solving a linear first-order differential equation.