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
Famous quotes containing the words order and/or equation:
“The theater, which is in no thing, but makes use of everything—gestures, sounds, words, screams, light, darkness—rediscovers itself at precisely the point where the mind requires a language to express its manifestations.... To break through language in order to touch life is to create or recreate the theatre.”
—Antonin Artaud (1896–1948)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)