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

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

Introductory Remarks

We now examine the general linear differential equation

\[\tag{23.1} f_n(x)y^{(n)} + f_{(n-1)}(x)y^{(n-1)} + \cdots + f_1(x)y' + f_0(x)y = Q(x),\]

and its related homogeneous equation

\[\tag{23.11} f_n(x)y^{(n)} + f_{(n-1)}(x)y^{(n-1)} + \cdots + f_1(x)y' + f_0(x)y = 0,\]

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 $f_n(x) \neq 0$ 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:

\[\tag{23.12} f_2(x)y'' + f_1(x)y' + f_0(x)y = Q(x),\]

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

\[\tag{23.14} f_2(x)y'' + f_1(x)y' + f_0(x)y = 0,\]

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:

\[\tag{23.15} f_2(x)y'' + f_1(x)y' + f_0(x)y = Q(x).\]

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

\[\tag{23.2} y_2(x) = y_1(x) \int{u(x)dx}\]

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 ${y_2}’$ 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:

\[\tag{23.3} y(x) = y_1(x) \int{u(x)dx}\]

See the book for details…