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Method of Variation of Parameters for Second-Order Linear Differential Equations with Constant Coefficients

This Demonstration shows how to solve a nonhomogeneous linear second-order differential equation of the form , where and are constants.
The corresponding homogeneous equation is with the characteristic equation . If and are two real roots of the characteristic equation, then the general solution of the homogeneous differential equation is , where and are arbitrary constants. If , the general solution is . If , the general solution is .
To find a particular solution of the nonhomogeneous equation, the method of variation of parameters (Lagrange's method) is used. The solution has the form , where and are independent partial solutions of the corresponding homogeneous equation, and and are functions of satisfying the system of equations
,
Define the Wronskian by . Then , .
The general solution of the nonhomogeneous equation is the sum of the general solution of the homogeneous equation and a particular solution of the nonhomogeneous equation.

DETAILS

The method of variation of parameters can be used even if and are not constants.
Reference
[1] V. P. Minorsky, Problems in Higher Mathematics, (Y. Ermolyev, trans.), Moscow: Mir Publishers, 1975 pp. 262–263.
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