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Reduction of order is a technique in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
for solving second-order linear ordinary
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. It is employed when one solution y_1(x) is known and a second
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
solution y_2(x) is desired. The method also applies to ''n''-th order equations. In this case the
ansatz In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
, 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 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
(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 In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian of ...
: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 In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calcul ...
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 In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible t ...


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 The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, place =
Providence Providence often refers to: * Providentia, the divine personification of foresight in ancient Roman religion * Divine providence, divinely ordained events and outcomes in Christianity * Providence, Rhode Island, the capital of Rhode Island in the ...
, 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