Second Order Homogeneous DE

A linear second order homogeneous differential equation involves terms up to the second derivative of a function. For the case of constant multipliers, The equation is of the form

and can be solved by the substitution

The solution which fits a specific physical situation is obtained by substituting the solution into the equation and evaluating the various constants by forcing the solution to fit the physical boundary conditions of the problem at hand. Substituting gives

which leads to a variety of solutions, depending on the values of a and b.

Applications

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Differential Equation Applications

Simple harmonic motion:

Simple pendulum:

Azimuthal equation, hydrogen atom:

Velocity profile in fluid flow.

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Applications of 1st Order Homogeneous Differential Equations

The general form of the solution of the homogeneous differential equation can be applied to a large number of physical problems.

Barometric pressure variation with altitude:

Discharge of a capacitor

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