Linear Differential Equation - First Order Equation

First Order Equation

Examples
Solve the equation

with the initial condition

Using the general solution method:

The indefinite integral is solved to give:

Then we can reduce to:

where κ is 4/3 from the initial condition.

A linear ODE of order 1 with variable coefficients has the general form

Where D is the differential operator. Equations of this form can be solved by multiplying the integrating factor

throughout to obtain

which simplifies due to the product rule to

which, on integrating both sides, yields

In other words: The solution of a first-order linear ODE

with coefficients that may or may not vary with x, is:

where is the constant of integration, and


A compact form of the general solution is (see J. Math. Chem. 48 (2010) 175):

where is the generalized Dirac delta function.

Read more about this topic:  Linear Differential Equation

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