Complex Math Simple Life

II Order Differential Equations with Constant Coefficients

Posted in differential, equation by beauangel on February 17, 2007

Standard Form

a_0 \frac{d^2y}{dx^2} + a_1 \frac{dy}{dx} + a_2y = f(x) where a_0, a_1, a_2 are constants.

The general solution of such a differential equation contains two parts; the Complementary Function and the Particular Integral.

To find the Complementary Function:

Write the auxiliary equation in the form: a_0m^2 + a_1m + a_2 =0 where the differential equation is in the standard form.

Solving the auxiliary equation, a quadratic, will yield two roots, say m1 and m2.

The Complementary Function (CF) is then written based on the roots.

  • If m1 and m2 are real and different, then the CF is y = Ae^{m_1x} + Be^{m_2x}
  • If m1 and m2 are real and equal, ie m1 = m2 = m, then the CF is y = (A + Bx)e^{mx}
  • If m1 and m2 are complex roots of the form \alpha + i\beta and \alpha - i\beta, then the CF is y = e^{\alpha x} (A \cos \beta x + B \sin \beta x) where A and B are arbitrary constants.

The Particular Integral is similar to the function of x, f(x) on the RHS of the differential equation and it is determined by the method of undetermined coefficients.

If the RHS of the equation is a constant, then we use a trial PI of K (a different constant). A trial PI of a general first degree form A_0x +A_1 and a general second degree form A_0x^2 + A_1x + A_2 is used for functions of ax + b and ax^2 + bx +c respectively. If the function is of the form be^{ax}, then we use Ke^{ax}, the same exponential function. Similarly, trigonometric functions will have trial PIs of the form L \cos bx + K \sin bx.

K, L, A_0, A_1, A_2 are undetermined coefficients. By finding \frac{dy}{dx} and \frac{d^2y}{dx^2} and substituting y, \frac{dy}{dx} and \frac{d^2y}{dx^2} into the given differential equation, the values of the undetermined coefficients can be obtained. Hence the Particular Integral is found.

Combine the Complementary Function and the Particular Integral and we will get the General Solution of the Second Order Differential Equation with constant coefficients.

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  1. Second Order Difference Equations « Complex Math Simple Life said,

    on February 25, 2007 on 10:35 am

    [...] The solving of a second order difference equation is very similar to the method of solving a second order differential equation, which is discussed in this previous post here. [...]


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