# variation of parameters

Suppose that we have an $n$th order linear differential operator  $L[y]:=y^{(n)}+p_{1}(t)y^{(n-1)}+\cdots+p_{n}(t)y,$

and a corresponding nonhomogeneous differential equation

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

Suppose that we know a fundamental set of solutions $y_{1},y_{2},\ldots,y_{n}$ of the corresponding homogeneous differential equation $L[y_{c}]=0$. The general solution of the homogeneous equation is

 $y_{c}(t)=c_{1}y_{1}(t)+c_{2}y_{2}(t)+\cdots+c_{n}y_{n}(t),$

where $c_{1},c_{2},\ldots,c_{n}$ 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 $c_{1},c_{2},\ldots,c_{n}$ 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

 $Y(t)=u_{1}(t)y_{1}(t)+u_{2}(t)y_{2}(t)+\cdots+u_{n}(t)y_{n}(t),$ (2)

where $u_{1}(t),u_{2}(t),\ldots,u_{n}(t)$ 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

 $\displaystyle y_{1}u_{1}^{\prime}+y_{2}u_{2}^{\prime}+\cdots+y_{n}u_{n}^{\prime}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle y_{1}^{\prime}u_{1}^{\prime}+y_{2}^{\prime}u_{2}^{\prime}+\cdots% +y_{n}^{\prime}u_{n}^{\prime}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle\vdots$ $\displaystyle y_{1}^{(n-2)}u_{1}^{\prime}+y_{2}^{(n-2)}u_{2}^{\prime}+\cdots+y% _{n}^{(n-2)}u_{n}^{\prime}$ $\displaystyle=$ $\displaystyle 0.$

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

 $y_{1}^{(n-1)}u_{1}^{\prime}+y_{2}^{(n-1)}u_{2}^{\prime}+\cdots+y_{n}^{(n-1)}u_% {n}^{\prime}=g.$

So we have a system of $n$ equations for $u_{1}^{\prime},u_{2}^{\prime},\ldots,u_{n}^{\prime}$ which we can solve using Cramer’s rule:

 $u_{m}^{\prime}(t)=\frac{g(t)W_{m}(t)}{W(t)},\quad m=1,2,\ldots,n.$

Such a solution always exists since the Wronskian $W=W(y_{1},y_{2},\ldots,y_{n})$ of the system is nowhere zero, because the $y_{1},y_{2},\ldots,y_{n}$ form a fundamental set of solutions. Lastly the term $W_{m}$ is the Wronskian determinant with the $m$th column replaced by the column $(0,0,\ldots,0,1)$.

Finally the particular solution can be written explicitly as

 $Y(t)=\sum_{m=1}^{n}y_{m}(t)\int\frac{g(t)W_{m}(t)}{W(t)}dt.$

## References

Title variation of parameters VariationOfParameters 2013-03-22 12:39:16 2013-03-22 12:39:16 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Theorem  msc 34A30 msc 34A05 variation of constants