Second Order Differential Equations

Second order differential equations involve the second derivative d^2y/dx^2 and are a major topic in AQA Further Mathematics. This lesson covers homogeneous and non-homogeneous linear equations with constant coefficients, the auxiliary equation method, and finding particular integrals.

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1 General Form

A second order linear ODE with constant coefficients has the form:

a (d^2y/dx^2) + b (dy/dx) + c y = f(x)

where a, b, c are constants and f(x) is a given function.

• If f(x) = 0, the equation is homogeneous.
• If f(x) is not zero, the equation is non-homogeneous (or inhomogeneous).

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2 The Auxiliary Equation (Homogeneous Case)

For a y'' + b y' + c y = 0, try the solution y = e^(mx). Substituting:

a m^2 e^(mx) + b m e^(mx) + c e^(mx) = 0

Dividing by e^(mx) (which is never zero):

a m^2 + b m + c = 0

This is the auxiliary equation (AE). The nature of its roots determines the form of the general solution.

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Case 1: Two Distinct Real Roots (m_1 and m_2)

y = A e^(m_1 x) + B e^(m_2 x)

where A and B are arbitrary constants.