variation of parameters

The method of variation of parameters is a way of finding a particularsolution 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 theconstants $c_1,c_2,\mathrm{\dots },c_n$ are chosen to satisfy the appropriateboundary conditions or initial conditions.

The key step in using variation of parameters is to suppose that theparticular 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 findthese $n$ 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 thefollowing 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 theabove 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 $u_1^{\prime },u_2^{\prime },\mathrm{\dots },u_n^{\prime }$ whichwe 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 afundamental set of solutions. Lastly the term $W_m$ is the Wronskiandeterminant with the $m$th column replaced by the column$(0,0,\mathrm{\dots },0,1)$.

Finally the particular solution can be written explicitly as

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