Linear Homogeneous Ordinary Differential Equations with Constant Coefficients
engineering fundamentals Linear Homogeneous
ODE with Constant Coefficients

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Standard Form

A linear homogeneous ordinary differential equation with constant coefficients has the general form of

where are all constants.


2nd Order Linear Homogeneous ODE with Constant Coefficients

A second order linear homogeneous ordinary differential equation with constant coefficients can be expressed as

This equation implies that the solution is a function whose derivatives keep the same form as the function itself and do not explicitly contain the independent variable , since constant coefficients are not capable of correcting any irregular formats or extra variables. An elementary function which satisfies this restriction is the exponential function .

Substitute the exponential function into the above differential equation, the characteristic equation of this differential equation is obtained

This characteristic equation has two roots and .

2nd Order Linear Homogeneous ODE with Constant Coefficients:
Characteristic Equation:
Solutions of Characteristic Equation , General Solution
1
2
3

nth Order Linear Homogeneous ODE with Constant Coefficients

Similar to the second order equations, the form, characteristic equation, and general solution of order linear homogeneous ordinary differential equations are summarized as follows:

nth Order Linear Homogeneous ODE with Constant Coefficients:
Characteristic Equation:
Solutions of Characteristic Equation General Solution
1 are all different real numbers.
2 are k repeated real roots; others are different real numbers.
3 are k/2 pairs of complex conjugate roots ; others are different real numbers.

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