Linear differential equation
In mathematics, a linear differential equation is a differential equation, most generally in the form
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(where D is the differential operator), is said to have order n.
To solve a linear differential equation one makes a substitution y=eλx in the homogeneous equation (ie., setting f(x)=0), to form the characteristic equation
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to obtain the solutions
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Where the solutions are distinct, we have immediately n solutions to the differential equation in the form
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and we have that the general solution to the homogeneous equation can be formed from a linear combination of the yi, ie.,
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Where the solutions are not distinct, it may be necessary to multiply them by some power of x to obtain linear dependence.
To obtain the solution to the inhomogeneous equation, find a particular solution yP(x) by the method of undetermined coefficients and the general solution to the linear differential equation is the sum of the homogeneous and the particular equation.
A linear differential equation can also refer to an equation in the form
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where this equation can be solved by forming the integrating factor  , multiplying throughout to obtain
which simplifies due to the product rule to
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on integrating both sides yields
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See also:
Referenced By
List of mathematical topics (J-L)
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