Solution of the Linear Differential Equation with Nonconstant Coefficients. Reduction of Order Method.

Introductory Remarks

We now examine the general linear differential equation

$$\begin{array}{cc}{f}_n(x)y^{(n)}+f_{(n-1)}(x)y^{(n-1)}+\cdots +f_1(x)y^{\prime }+f_0(x)y=Q(x),& \text{(23.1)}\end{array}$$

and its related homogeneous equation

$$\begin{array}{cc}{f}_n(x)y^{(n)}+f_{(n-1)}(x)y^{(n-1)}+\cdots +f_1(x)y^{\prime }+f_0(x)y=0,& \text{(23.11)}\end{array}$$

where $f_0(x),f_1(x),\cdots ,f_n(x),Q(x)$ are each continuous functions of $x$ on a common interval $I$ and $f_n(x)\ne 0$ when $x$ is in $I$.

There are no standard methods of finding solutions of $(23.1)$ in the general case. For an unrestricted $(23.11)$ that has a solution expressible in terms of elementary functions, you can use a standard method to find one independent solution, if the other $n-1$ independent solutions are known. For $(23.1)$, the best you can obtain from a standard method is to find one independent solution of $(23.11)$ and a particular solution of $(23.1)$, provided the other $n-1$ independent solutions of $(23.11)$ are known.

Therefore, even when following a standard method for finding a general solution of only the second order equation:

$$\begin{array}{cc}{f}_2(x)y^{″}+f_1(x)y^{\prime }+f_0(x)y=Q(x),& \text{(23.12)}\end{array}$$

it is essential that $f_0(x),f_1(x),f_2(x)$ be of such a character that the needed first solution of its related homogenous equation can be discovered.

Reduction of Order Method

Assuming we have been able to find a non-trivial solution $y_1$ (a solution other than $y\equiv 0$, which is always true) to the homogenous equation

$$\begin{array}{cc}{f}_2(x)y^{″}+f_1(x)y^{\prime }+f_0(x)y=0,& \text{(23.14)}\end{array}$$

There is a method, called the reduction of order method, to obtain a second independent solution of $(23.14)$ as well as a particular solution of the related nonhomogenous equation:

$$\begin{array}{cc}{f}_2(x)y^{″}+f_1(x)y^{\prime }+f_0(x)y=Q(x).& \text{(23.15)}\end{array}$$

Let $y_2(x)$ be a second solution of (23.14) and assume that it will have the form:

$$\begin{array}{cc}{y}_2(x)=y_1(x)\int u(x)dx& \text{(23.2)}\end{array}$$

where $u(x)$ is an unknown function to be determined.

(jmh why do we/why can we make this assumption? the book does not explain).

We can then find ${y_2}^{\prime }$ and ${y_2}^{″}$ via differntiation and substitute them into $(23.14)$ and simplify, which will result in a first order differential equation of $u$. We can then solve for $u$ and substitute it back into $(23.2)$ to solve for $y_2$.

Now that we know $y_1$ and $y_2$, we can write the general solution of $(23.14)$ as $y=c_1{y}_1+c_2{y}_2$.

The same approach can be used for finding a general solution to $(23.15)$ using the substitution:

$$\begin{array}{cc}y(x)=y_1(x)\int u(x)dx& \text{(23.3)}\end{array}$$

See the book for details...