Method of variation of parameters


Method of variation of parameters

In mathematics, variation of parameters also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. It was developed by the Italian-French mathematician Joseph Louis Lagrange.

For first-order inhomogeneous linear differential equations it's usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods are rather heuristics that involve guessing and don't work for all inhomogenous linear differential equations.

Given an ordinary non-homogeneous linear differential equation of order "n":y^{(n)}(x) + sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = b(x). (i)let y_1(x), ldots, y_n(x) be a fundamental system of the corresponding homogeneous equation:y^{(n)}(x) + sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = 0. (ii)

Then a particular solution to the non-homogeneous equation is given by:y_p(x) = sum_{i=1}^{n} c_i(x) y_i(x) (iii)where the c_i(x) are continuous functions which satisfy the equations:sum_{i=1}^{n} c_i^'(x) y_i^{(j)}(x) = 0 , mathrm{,} quad j = 0,ldots, n-2 (iv)

(results from substitution of (iii) into the homogeneous case (ii); )
and:sum_{i=1}^{n} c_i^'(x) y_i^{(n-1)}(x) = b(x).. (v)
(results from substitution of (iii) into (i) and applying (iv);

c_i'(x)=0 for all x and i is the only way to satisfy the condition, since all y_i(x) are linearly independent. It implies that all c_i(x) are independent of x in the homogeneous case b(x)=0. )

This linear system of "n" equations can then be solved using Cramer's rule yielding:c_i^'(x) = frac{W_i(x)}{W(x)} , mathrm{,} quad i=1,ldots,nwhere W(x) is the Wronskian determinant of the fundamental system and W_i(x) is the Wronskian determinant of the fundamental system with the "i"-th column replaced by (0, 0, ldots, b(x)).

The particular solution to the non-homogeneous equation can then be written as:sum_{i=1}^n int frac{W_i(x)}{W(x)} dx , y_i(x).

Examples

Specific second order equation

Let us solve : y"+4y'+4y=cosh{x}.;!

We want to find the general solution to the differential equation, that is, we want to find solutions to the homogeneous differential equation: y"+4y'+4y=0.;!Form the characteristic equation: lambda^2+4lambda+4=(lambda+2)^2=0;!: lambda=-2,-2.;!Since we have a repeated root, we have to introduce a factor of "x" for one solution to ensure linear independence. So, we obtain "u"1="e"-2"x", and "u"2="xe"-2"x". The Wronskian of these two functions is : egin{vmatrix} e^{-2x} & xe^{-2x} \-2e^{-2x} & -e^{-2x}(2x-1)\end{vmatrix} = -e^{-2x}e^{-2x}(2x-1)+2xe^{-2x}e^{-2x} := -e^{-4x}(2x-1)+2xe^{-4x}= (-2x+1+2x)e^{-4x} = e^{-4x}.;!

Because the Wronskian is non-zero, the two functions are linearly independent, so this is in fact the general solution for the homogeneous differential equation (and not a mere subset of it).

We seek functions "A"("x") and "B"("x") so "A"("x")"u"1+"B"("x")"u"2 is a general solution of the inhomogeneous equation. We need only calculate the integrals:A(x) = - int {1over W} u_2(x) f(x),dx,; B(x) = int {1 over W} u_1(x)f(x),dxthat is,:A(x) = - int {1over e^{-4x xe^{-2x} cosh{x},dx = - int xe^{2x}cosh{x},dx = -{1over 18}e^x(9(x-1)+e^{2x}(3x-1))+C_1:B(x) = int {1 over e^{-4x e^{-2x} cosh{x},dx = int e^{2x}cosh{x},dx ={1over 6}e^{x}(3+e^{2x})+C_2 where C_1 and C_2 are constants of integration.

General second order equation

We have a differential equation of the form:u"+p(x)u'+q(x)u=f(x),and we define the linear operator :L=D^2+p(x)D+q(x),where "D" represents the differential operator. We therefore have to solve the equation L u(x)=f(x) for u(x), where L and f(x) are known.

We must solve first the corresponding homogeneous equation::u"+p(x)u'+q(x)u=0,by the technique of our choice. Once we've obtained two linearly independent solutions to this homogeneous differential equation (because this ODE is second-order) — call them "u"1 and "u"2 — we can proceed with variation of parameters.

Now, we seek the general solution to the differential equation u_G(x) which we assume to be of the form:u_G(x)=A(x)u_1(x)+B(x)u_2(x).,

Here, A(x) and B(x) are unknown and u_1(x) and u_2(x) are the solutions to the homogeneous equation. Observe that if A(x) and B(x) are constants, then Lu_G(x)=0. We desire "A"="A"("x") and "B"="B"("x") to be of the form :A'(x)u_1(x)+B'(x)u_2(x)=0.,

Now,:u_G'(x)=(A(x)u_1(x)+B(x)u_2(x))'=(A(x)u_1(x))'+(B(x)u_2(x))',:=A'(x)u_1(x)+A(x)u_1'(x)+B'(x)u_2(x)+B(x)u_2'(x),:=A'(x)u_1(x)+B'(x)u_2(x)+A(x)u_1'(x)+B(x)u_2'(x),and since we have required the above condition, then we have :u_G'(x)=A(x)u_1'(x)+B(x)u_2'(x).,Differentiating again (omitting intermediary steps):u_G"(x)=A(x)u_1"(x)+B(x)u_2"(x)+A'(x)u_1'(x)+B'(x)u_2'(x).,

Now we can write the action of "L" upon "u""G" as:Lu_G=A(x)Lu_1(x)+B(x)Lu_2(x)+A'(x)u_1'(x)+B'(x)u_2'(x).,Since "u"1 and "u"2 are solutions, then:Lu_G=A'(x)u_1'(x)+B'(x)u_2'(x).,

We have the system of equations:egin{pmatrix}u_1(x) & u_2(x) \u_1'(x) & u_2'(x) end{pmatrix}egin{pmatrix}A'(x) \B'(x)end{pmatrix} =egin{pmatrix}0\fend{pmatrix}.Expanding, :egin{pmatrix}A'(x)u_1(x)+B'(x)u_2(x)\A'(x)u_1'(x)+B'(x)u_2'(x)end{pmatrix} =egin{pmatrix}0\fend{pmatrix}.So the above system determines precisely the conditions:A'(x)u_1(x)+B'(x)u_2(x)=0,:A'(x)u_1'(x)+B'(x)u_2'(x)=Lu_G=f.,

We seek "A"("x") and "B"("x") from these conditions, so, given:egin{pmatrix}u_1(x) & u_2(x) \u_1'(x) & u_2'(x) end{pmatrix}egin{pmatrix}A'(x) \B'(x)end{pmatrix} =egin{pmatrix}0\fend{pmatrix}we can solve for ("A"′("x"), "B"′("x"))"T", so:egin{pmatrix}A'(x) \B'(x)end{pmatrix}=egin{pmatrix}u_1(x) & u_2(x) \u_1'(x) & u_2'(x) end{pmatrix}^{-1}egin{pmatrix}0\fend{pmatrix}:={1over W}egin{pmatrix}u_2'(x) & -u_2(x) \-u_1'(x) & u_1(x) end{pmatrix}egin{pmatrix}0\fend{pmatrix},where "W" denotes the Wronskian of "u"1 and "u"2. (We know that "W" is nonzero, from the assumption that "u"1 and "u"2 are linearly independent.)

So,:A'(x) = - {1over W} u_2(x) f(x),; B'(x) = {1 over W} u_1(x)f(x):A(x) = - int {1over W} u_2(x) f(x),dx,; B(x) = int {1 over W} u_1(x)f(x),dx.

Whilst homogeneous equations are relatively easy to solve, this method allows the calculation of the coefficients of the general solution of the "in"homogeneous equation, and thus the complete general solution of the inhomogeneous equation can be determined.

Note that A(x) and B(x) are each determined only up to an arbitrary additive constant (the constant of integration); one would expect two constants of integration because the original equation was second order. Adding a constant to A(x) or B(x) does not change the value of Lu_G(x) because L is linear.

=Literature="Elementary Differntial Equations and Boundary Value problems ", W.E. Boyce and R.C. DiPrima, Wiley Interscience, 1965


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Variation of parameters — In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. It was developed by Joseph Louis Lagrange[citation needed]. For first order… …   Wikipedia

  • variation of parameters — a method for solving a differential equation by first solving a simpler equation and then generalizing this solution properly so as to satisfy the original equation by treating the arbitrary constants not as constants but as variables …   Useful english dictionary

  • Method of undetermined coefficients — In mathematics, the method of undetermined coefficients, also known as the lucky guess method, is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations. It is closely… …   Wikipedia

  • parameters, variation of — ▪ mathematics       general method for finding a particular solution of a differential equation by replacing the constants in the solution of a related (homogeneous) equation by functions and determining these functions so that the original… …   Universalium

  • Method overriding — Method overriding, in object oriented programming, is a language feature that allows a subclass or child class to provide a specific implementation of a method that is already provided by one of its superclasses or parent classes. The… …   Wikipedia

  • Method of moments (statistics) — See method of moments (probability theory) for an account of a technique for proving convergence in distribution. In statistics, the method of moments is a method of estimation of population parameters such as mean, variance, median, etc. (which… …   Wikipedia

  • Variational method (quantum mechanics) — The variational method is, in quantum mechanics, one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. The basis for this method is the variational principle. Introduction Suppose we are given …   Wikipedia

  • Annihilator method — In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of inhomogeneous ordinary differential equations. It is equivalent to the method of undetermined coefficients, and the two names are… …   Wikipedia

  • Stein's method — is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it 1972,[1] to obtain a bound between… …   Wikipedia

  • Reassignment method — The method of reassignment is a technique forsharpening a time frequency representation by mappingthe data to time frequency coordinates that are nearer tothe true region of support of theanalyzed signal. The method has been… …   Wikipedia