Method Of Variation Of Parameters Cases

The above procedure is what we call the method of variation of parameters for solving a second-order nonhomogeneous differential equation. Notice the similarity between the two equationsinthesystem.

21 Example Two Methods Solve y y ex by undetermined coefficients and by variation of parameters. Explain any differences in the answers. Solution The general solution is reported to be y yh yp c1ex c2ex xex2. Details follow. Homogeneous solution. The characteristic equation r2 1 0 for yy 0 has roots 1.

4.6.2 Lagrange's Method of Variation of Parameters Suppose that two independent solutions y1 and y2 of the homogeneous linear equation Ly aty00 bty0 cty 0 are known. Then of course the function v1y1 v2y2 is also a solution if v1 and v2 are constants.

On top of that undetermined coefficients will only work for a fairly small class of functions. The method of Variation of Parameters is a much more general method that can be used in many more cases. However, there are two disadvantages to the method. First, the complementary solution is absolutely required to do the problem.

2. Variation of Parameters the method of reduction of order. The latter asks whether it is possible to multiply a known homoge-neous solution by some function to obtain another independent homogeneous solution. The former variation of parameters supposes that a fundamental set of solutions is say, y00 pxy0 qxy 0

The method of variation of parameters can be applied to all linear differential equations. It is therefore more powerful than the method of undetermined coefficients, which is restricted to linear differential equations with constant coefficients and particular forms of 92phi 92left x92right x.

Variation of Parameters that we will learn here which works on a wide range of functions but is a little messy to use. Variation of Parameters To keep things simple, we are only going to look at the case d2y dx2 p dy dx qy f x where p and q are constants and f x is a non-zero function of x.

Section 4.7 Variations of parameters The method of variation of parameters also called method of variation of constants or method of Lagrange is a method for finding a particular solution of systems of first-order linear differential equations x0 Ptx gt second order nonhomogeneous linear differential equations y00 pty0 qty

An intelligent guess, based upon the Method of Undetermined Coefficients, was reviewed previously in Chapter 1. However, a more methodical method, which is first seen in a first course in differential equations, is the Method of Variation of Parameters.

The method of variation of parameters is another technique used to find particular solutions to nonhomogeneous linear differential equations. It is especially useful for equations with both constant and variable coefficients and is applicable when the forcing function, f x, makes the method of undetermined coefficients impractical.