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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