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

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This Demonstration shows how to solve a nonhomogeneous linear second-order differential equation of the form , where and are constants.

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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.

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Contributed by: Izidor Hafner (April 2014)
Open content licensed under CC BY-NC-SA


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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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