Differential Equations - Higher Order: Variation of Parameters - i

We now need to take a look at the second method of determining a particular solution to a differential equation. As we did when we first saw Variation of Parameters we’ll go through the whole process and derive up a set of formulas that can be used to generate a particular solution.

However, as we saw previously when looking at 2^nd order differential equations this method can lead to integrals that are not easy to evaluate. So, while this method can always be used, unlike Undetermined Coefficients, to at least write down a formula for a particular solution it is not always going to be possible to actually get a solution.

So let’s get started on the process. We’ll start with the differential equation,

$$\begin{array}{}{y}^{\left(n\right)}+p_{n-1}\left(t\right)y^{\left(n-1\right)}+\cdots +p_1\left(t\right)y^{\prime }+p_0\left(t\right)y=g\left(t\right)& \text{(1)}\end{array}$$

and assume that we’ve found a fundamental set of solutions, $y_1\left(t\right),y_2\left(t\right),\dots ,y_n\left(t\right)$, for the associated homogeneous differential equation.

Because we have a fundamental set of solutions to the homogeneous differential equation we now know that the complementary solution is,

$$y\left(t\right)=c_1{y}_1\left(t\right)+c_2{y}_2\left(t\right)+\cdots +c_n{y}_n\left(t\right)$$

The method of variation of parameters involves trying to find a set of new functions, $u_1\left(t\right),u_2\left(t\right),\dots ,u_n\left(t\right)$ so that,

$$\begin{array}{}Y\left(t\right)=u_1\left(t\right)y_1\left(t\right)+u_2\left(t\right)y_2\left(t\right)+\cdots +u_n\left(t\right)y_n\left(t\right)& \text{(2)}\end{array}$$

will be a solution to the nonhomogeneous differential equation. In order to determine if this is possible, and to find the $u_i\left(t\right)$ if it is possible, we’ll need a total of $n$ equations involving the unknown functions that we can (hopefully) solve.

One of the equations is easy. The guess, $\text{(2)}$, will need to satisfy the original differential equation, $\text{(1)}$. So, let’s start taking some derivatives and as we did when we first looked at variation of parameters we’ll make some assumptions along the way that will simplify our work and in the process generate the remaining equations we’ll need.

The first derivative of $\text{(2)}$ is,

$$Y^{\prime }\left(t\right)=u_1{y}_1^{\prime }+u_2{y}_2^{\prime }+\cdots +u_n{y}_n^{\prime }+u_1^{\prime }{y}_1+u_2^{\prime }{y}_2+\cdots +u_n^{\prime }{y}_n$$

Note that we rearranged the results of the differentiation process a little here and we dropped the $\left(t\right)$ part on the $u$ and $y$ to make this a little easier to read. Now, if we keep differentiating this it will quickly become unwieldy and so let’s make as assumption to simplify things here. Because we are after the $u_i\left(t\right)$ we should probably try to avoid letting the derivatives on these become too large. So, let’s make the assumption that,

$$u_1^{\prime }{y}_1+u_2^{\prime }{y}_2+\cdots +u_n^{\prime }{y}_n=0$$

The natural question at this point is does this even make sense to do? The answer is, if we end up with a system of $n$equations that we can solve for the $u_i\left(t\right)$ then yes it does make sense to do. Of course, the other answer is, we wouldn’t be making this assumption if we didn’t know that it was going to work. However, to accept this answer requires that you trust us to make the correct assumptions so maybe the first answer is the best at this point.

At this point the first derivative of $\text{(2)}$ is,

$$Y^{\prime }\left(t\right)=u_1{y}_1^{\prime }+u_2{y}_2^{\prime }+\cdots +u_n{y}_n^{\prime }$$

and so we can now take the second derivative to get,

$$Y^{\prime \prime }\left(t\right)=u_1{y}_1^{\prime \prime }+u_2{y}_2^{\prime \prime }+\cdots +u_n{y}_n^{\prime \prime }+u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }+\cdots +u_n^{\prime }{y}_n^{\prime }$$

This looks an awful lot like the original first derivative prior to us simplifying it so let’s again make a simplification. We’ll again want to keep the derivatives on the $u_i\left(t\right)$ to a minimum so this time let’s assume that,

$$u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }+\cdots +u_n^{\prime }{y}_n^{\prime }=0$$

and with this assumption the second derivative becomes,

$$Y^{\prime \prime }\left(t\right)=u_1{y}_1^{\prime \prime }+u_2{y}_2^{\prime \prime }+\cdots +u_n{y}_n^{\prime \prime }$$

Hopefully you’re starting to see a pattern develop here. If we continue this process for the first $n-1$ derivatives we will arrive at the following formula for these derivatives.

$$\begin{array}{}{Y}^{\left(k\right)}\left(t\right)=u_1{y}_1^{\left(k\right)}+u_2{y}_2^{\left(k\right)}+\cdots +u_n{y}_n^{\left(k\right)}k=1,2,\dots ,n-1& \text{(3)}\end{array}$$

To get to each of these formulas we also had to assume that,

$$\begin{array}{}{u}_1^{\prime }{y}_1^{\left(k\right)}+u_2^{\prime }{y}_2^{\left(k\right)}+\cdots +u_n^{\prime }{y}_n^{\left(k\right)}=0k=0,1,\dots n-2& \text{(4)}\end{array}$$

and recall that the 0^th derivative of a function is just the function itself. So, for example, $y_2^{\left(0\right)}\left(t\right)=y_2\left(t\right)$.

Notice as well that the set of assumptions in $\text{(4)}$ actually give us $n-1$ equations in terms of the derivatives of the unknown functions : $u_1\left(t\right),u_2\left(t\right),\dots ,u_n\left(t\right)$.

All we need to do then is finish generating the first equation we started this process to find (i.e. plugging $\text{(2)}$ into $\text{(1)}$). To do this we’ll need one more derivative of the guess. Differentiating the ${\left(n-1\right)}^{\text{st}}$ derivative, which we can get from $\text{(3)}$, to get the $n$^th derivative gives,

$$Y^{\left(n\right)}\left(t\right)=u_1{y}_1^{\left(n\right)}+u_2{y}_2^{\left(n\right)}+\cdots +u_n{y}_n^{\left(n\right)}+u_1^{\prime }{y}_1^{\left(n-1\right)}+u_2^{\prime }{y}_2^{\left(n-1\right)}+\cdots +u_n^{\prime }{y}_n^{\left(n-1\right)}$$

This time we’ll also not be making any assumptions to simplify this but instead just plug this along with the derivatives given in $\text{(3)}$ into the differential equation, $\text{(1)}$

$$\begin{array}{cc}{u}_1{y}_1^{\left(n\right)}+u_2{y}_2^{\left(n\right)}+\cdots +u_n{y}_n^{\left(n\right)}+{u^{\prime }}_1{y}_1^{\left(n-1\right)}+{u^{\prime }}_2{y}_2^{\left(n-1\right)}+\cdots +{u^{\prime }}_n{y}_n^{\left(n-1\right)}& +\\ p_{n-1}\left(t\right)\left[u_1{y}_1^{\left(n-1\right)}+u_2{y}_2^{\left(n-1\right)}+\cdots +u_n{y}_n^{\left(n-1\right)}\right]& +\\ ⋮& \\ p_1\left(t\right)\left[u_1{y^{\prime }}_1+u_2{y^{\prime }}_2+\cdots +u_n{y^{\prime }}_n\right]& +\\ p_0\left(t\right)\left[u_1{y}_1+u_2{y}_2+\cdots +u_n{y}_n\right]& =g\left(t\right)\end{array}$$

Next, rearrange this a little to get,

$$\begin{array}{cc}{u}_1\left[y_1^{\left(n\right)}+p_{n-1}\left(t\right)y_1^{\left(n-1\right)}+\cdots +p_1\left(t\right){y^{\prime }}_1+p_0\left(t\right)y_1\right]& +\\ u_2\left[y_2^{\left(n\right)}+p_{n-1}\left(t\right)y_2^{\left(n-1\right)}+\cdots +p_1\left(t\right){y^{\prime }}_2+p_0\left(t\right)y_2\right]& +\\ ⋮& \\ u_n\left[y_n^{\left(n\right)}+p_{n-1}\left(t\right)y_n^{\left(n-1\right)}+\cdots +p_1\left(t\right){y^{\prime }}_n+p_0\left(t\right)y_n\right]& +\\ {u^{\prime }}_1{y}_1^{\left(n-1\right)}+{u^{\prime }}_2{y}_2^{\left(n-1\right)}+\cdots +{u^{\prime }}_n{y}_n^{\left(n-1\right)}& =g\left(t\right)\end{array}$$

Recall that $y_1\left(t\right),y_2\left(t\right),\dots ,y_n\left(t\right)$ are all solutions to the homogeneous differential equation and so all the quantities in the $\left[\right]$ are zero and this reduces down to,

$$u_1^{\prime }{y}_1^{\left(n-1\right)}+u_2^{\prime }{y}_2^{\left(n-1\right)}+\cdots +u_n^{\prime }{y}_n^{\left(n-1\right)}=g\left(t\right)$$

So, this equation, along with those given in $\text{(4)}$, give us the $n$ equations that we needed. Let’s list them all out here for the sake of completeness.

$$\begin{array}{cc}{u^{\prime }}_1{y}_1+{u^{\prime }}_2{y}_2+\cdots +{u^{\prime }}_n{y}_n& =0\\ {u^{\prime }}_1{y^{\prime }}_1+{u^{\prime }}_2{y^{\prime }}_2+\cdots +{u^{\prime }}_n{y^{\prime }}_n& =0\\ {u^{\prime }}_1{y^{\prime \prime }}_1+{u^{\prime }}_2{y^{\prime \prime }}_2+\cdots +{u^{\prime }}_n{y^{\prime \prime }}_n& =0\\ ⋮& \\ {u^{\prime }}_1{y}_1^{\left(n-2\right)}+{u^{\prime }}_2{y}_2^{\left(n-2\right)}+\cdots +{u^{\prime }}_n{y}_n^{\left(n-2\right)}& =0\\ {u^{\prime }}_1{y}_1^{\left(n-1\right)}+{u^{\prime }}_2{y}_2^{\left(n-1\right)}+\cdots +{u^{\prime }}_n{y}_n^{\left(n-1\right)}& =g\left(t\right)\end{array}$$