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Linear Second-Order Homogeneous Equations
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General form of a second-order homogeneous differential equation
Solving involves finding the roots of the characteristic equation, which can be real and distinct, repeated, or complex. The general solution will depend on the nature of the roots.
Characteristic equation with real and distinct roots
The solution is the sum of two exponential functions:
Characteristic equation with repeated roots
The solution is an exponential function times a polynomial:
Characteristic equation with complex roots
The solution involves trigonometric functions:
Auxiliary or Characteristic Equation
The characteristic equation of a second-order homogeneous differential equation is
Reduction of Order
A technique used to find the second solution of a linear second-order differential equation when one solution is known: Assume the second solution has the form
Homogeneous Equations with Variable Coefficients
These differential equations have no general method for solution but can sometimes be solved using power series, the method of Frobenius, or by recognizing a special form.
Superposition Principle
For linear homogeneous differential equations, the sum of two solutions is also a solution. Therefore, the general solution can be expressed as a linear combination of independent solutions.
Wronskian and Linear Independence
The Wronskian determinant of two solutions of a second-order differential equation can be used to test for their linear independence, which is required for the general solution to span all solutions.
Cauchy-Euler Equations
A type of variable coefficient homogeneous second-order differential equation that can be transformed into a constant coefficient equation:
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