variation of parameters

The method of variation of parameters is a way of finding a particular solution to a nonhomogeneous linear differential equation.

Suppose that we have an $n$th order linear differential operator

$$L[y]:=y^{(n)}+p_1(t)y^{(n-1)}+\mathrm{\cdots }+p_n(t)y,$$

and a corresponding nonhomogeneous differential equation

$$L[y]=g(t).$$ (1)

Suppose that we know a fundamental set of solutions $y_1,y_2,\mathrm{\dots },y_n$ of the corresponding homogeneous differential equation $L[y_c]=0$. The general solution of the homogeneous equation is

$$y_c(t)=c_1{y}_1(t)+c_2{y}_2(t)+\mathrm{\cdots }+c_n{y}_n(t),$$

where $c_1,c_2,\mathrm{\dots },c_n$ are constants. The general solution to the nonhomogeneous equation $L[y]=g(t)$ is then

$$y(t)=y_c(t)+Y(t),$$

where $Y(t)$ is a particular solution which satisfies $L[Y]=g(t)$, and the constants $c_1,c_2,\mathrm{\dots },c_n$ are chosen to satisfy the appropriate boundary conditions or initial conditions.

The key step in using variation of parameters is to suppose that the particular solution is given by

$$Y(t)=u_1(t)y_1(t)+u_2(t)y_2(t)+\mathrm{\cdots }+u_n(t)y_n(t),$$ (2)

where $u_1(t),u_2(t),\mathrm{\dots },u_n(t)$ are as yet to be determined functions (hence the name variation of parameters). To find these $n$ functions we need a set of $n$ independent equations. One obvious condition is that the proposed ansatz satisfies Eq. (1). Many possible additional conditions are possible, we choose the ones that make further calculations easier. Consider the following set of $n-1$ conditions

$y_1{u}_1^{\prime }+y_2{u}_2^{\prime }+\mathrm{\cdots }+y_n{u}_n^{\prime }$ $=$ $0$

$y_1^{\prime }{u}_1^{\prime }+y_2^{\prime }{u}_2^{\prime }+\mathrm{\cdots }+y_n^{\prime }{u}_n^{\prime }$ $=$ $0$

$\mathrm{⋮}$

$y_1^{(n-2)}{u}_1^{\prime }+y_2^{(n-2)}{u}_2^{\prime }+\mathrm{\cdots }+y_n^{(n-2)}{u}_n^{\prime }$ $=$ $0.$

Now, substituting Eq. (2) into $L[Y]=g(t)$ and using the above conditions, we can get another equation

$$y_1^{(n-1)}{u}_1^{\prime }+y_2^{(n-1)}{u}_2^{\prime }+\mathrm{\cdots }+y_n^{(n-1)}{u}_n^{\prime }=g.$$

So we have a system of $n$ equations for $u_1^{\prime },u_2^{\prime },\mathrm{\dots },u_n^{\prime }$ which we can solve using Cramer’s rule:

$$u_m^{\prime }(t)=\frac{g(t)W_m(t)}{W(t)},m=1,2,\mathrm{\dots },n.$$

Such a solution always exists since the Wronskian $W=W(y_1,y_2,\mathrm{\dots },y_n)$ of the system is nowhere zero, because the $y_1,y_2,\mathrm{\dots },y_n$ form a fundamental set of solutions. Lastly the term $W_m$ is the Wronskian determinant with the $m$th column replaced by the column $(0,0,\mathrm{\dots },0,1)$.

Finally the particular solution can be written explicitly as

$$Y(t)=\sum _{m=1}^n{y}_m(t)\int \frac{g(t)W_m(t)}{W(t)}𝑑t.$$

Title	variation of parameters
Canonical name	VariationOfParameters
Date of creation	2013-03-22 12:39:16
Last modified on	2013-03-22 12:39:16
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	8
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 34A30
Classification	msc 34A05
Synonym	variation of constants