Reduction Of Order

Reduction of order is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution $y_1(x)$ is known and a second linearly independent solution $y_2(x)$ is desired. The method also applies to ''n''-th order equations. In this case the ansatz will yield an (''n''−1)-th order equation for $v$.

Second-order linear ordinary differential equations

An example

Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE)

:$a y''(x) + b y'(x) + c y(x) = 0,$

where $a, b, c$ are real non-zero coefficients. Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, $b^2 - 4 a c$, vanishes. In this case,

:$a y''(x) + b y'(x) + \frac y(x) = 0,$

from which only one solution,

:$y_1(x) = e^,$

can be found using its characteristic equation.

The method of reduction of order is used to obtain a second linearly independent solution to this differential equation using our one known solution. To find a second solution we take as a guess

:$y_2(x) = v(x) y_1(x)$

where $v(x)$ is an unknown function to be determined. Since $y_2(x)$ must satisfy the original ODE, we substitute it back in to get

:$a \left( v'' y_1 + 2 v' y_1' + v y_1'' \right) + b \left( v' y_1 + v y_1' \right) + \frac v y_1 = 0.$

Rearranging this equation in terms of the derivatives of $v(x)$ we get

:$\left(a y_1 \right) v'' + \left( 2 a y_1' + b y_1 \right) v' + \left( a y_1'' + b y_1' + \frac y_1 \right) v = 0.$

Since we know that $y_1(x)$ is a solution to the original problem, the coefficient of the last term is equal to zero. Furthermore, substituting $y_1(x)$ into the second term's coefficient yields (for that coefficient)

:$2 a \left( - \frac e^ \right) + b e^ = \left( -b + b \right) e^ = 0.$

Therefore, we are left with

:$a y_1 v'' = 0.$

Since $a$ is assumed non-zero and $y_1(x)$ is an exponential function (and thus always non-zero), we have

:$v'' = 0.$

This can be integrated twice to yield

:$v(x) = c_1 x + c_2$

where $c_1, c_2$ are constants of integration. We now can write our second solution as

:$y_2(x) = ( c_1 x + c_2 ) y_1(x) = c_1 x y_1(x) + c_2 y_1(x).$

Since the second term in $y_2(x)$ is a scalar multiple of the first solution (and thus linearly dependent) we can drop that term, yielding a final solution of

:$y_2(x) = x y_1(x) = x e^.$

Finally, we can prove that the second solution $y_2(x)$ found via this method is linearly independent of the first solution by calculating the Wronskian

:$W(y_1,y_2)(x) = \begin y_1 & x y_1 \\ y_1' & y_1 + x y_1' \end = y_1 ( y_1 + x y_1' ) - x y_1 y_1' = y_1^ + x y_1 y_1' - x y_1 y_1' = y_1^ = e^ \neq 0.$

Thus $y_2(x)$ is the second linearly independent solution we were looking for.

General method

Given the general non-homogeneous linear differential equation

:$y'' + p(t)y' + q(t)y = r(t)$

and a single solution $y_1(t)$ of the homogeneous equation math>r(t)=0 let us try a solution of the full non-homogeneous equation in the form:

:$y_2 = v(t)y_1(t)$

where $v(t)$ is an arbitrary function. Thus

:$y_2' = v'(t)y_1(t) + v(t)y_1'(t)$

and

:$y_2'' = v''(t)y_1(t) + 2v'(t)y_1'(t) + v(t)y_1''(t).$

If these are substituted for $y$, $y'$, and $y''$ in the differential equation, then

:$y_1(t)\,v'' + (2y_1'(t)+p(t)y_1(t))\,v' + (y_1''(t)+p(t)y_1'(t)+q(t)y_1(t))\,v = r(t).$

Since $y_1(t)$ is a solution of the original homogeneous differential equation, $y_1''(t)+p(t)y_1'(t)+q(t)y_1(t)=0$, so we can reduce to

:$y_1(t)\,v'' + (2y_1'(t)+p(t)y_1(t))\,v' = r(t)$

which is a first-order differential equation for $v'(t)$ (reduction of order). Divide by $y_1(t)$, obtaining

:$v''+\left(\frac+p(t)\right)\,v'=\frac.$

The integrating factor is $\mu(t)=e^=y_1^2(t)e^$.

Multiplying the differential equation by the integrating factor $\mu(t)$, the equation for $v(t)$ can be reduced to
:$\frac\left(v'(t) y_1^2(t) e^\right) = y_1(t)r(t)e^.$

After integrating the last equation, $v'(t)$ is found, containing one constant of integration. Then, integrate $v'(t)$ to find the full solution of the original non-homogeneous second-order equation, exhibiting two constants of integration as it should:

:$y_2(t) = v(t)y_1(t).$

See also

* Variation of parameters

References

* W. E. Boyce and R. C. DiPrima, ''Elementary Differential Equations and Boundary Value Problems (8th edition)'', John Wiley & Sons, Inc., 2005. .
* {{cite book
, last = Teschl
, given = Gerald
, authorlink=Gerald Teschl
, title = Ordinary Differential Equations and Dynamical Systems
, publisher=American Mathematical Society
, place = Providence
, year = 2012
, isbn = 978-0-8218-8328-0
, url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/
* Eric W. Weisstein,
Second-Order Ordinary Differential Equation Second Solution
', From MathWorld—A Wolfram Web Resource.

Ordinary differential equations