Reduction of order

Short description: Technique for solving linear ordinary differential equations

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Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y1(x) is known and a second linearly independent solution y2(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.

## Contents

* 1 Second-order linear ordinary differential equations
* 1.1 An example
* 1.2 General method
* 2 See also
* 3 References

## Second-order linear ordinary differential equations

### An example

Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE) ay″(x)+by′(x)+cy(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, b2−4ac, vanishes. In this case, ay″(x)+by′(x)+b24ay(x)=0, from which only one solution, y1(x)=e−b2ax, 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 y2(x)=v(x)y1(x) where v(x) is an unknown function to be determined. Since y2(x) must satisfy the original ODE, we substitute it back in to get a(v″y1+2v′y1′+vy1″)+b(v′y1+vy1′)+b24avy1=0. Rearranging this equation in terms of the derivatives of v(x) we get (ay1)v″+(2ay1′+by1)v′+(ay1″+by1′+b24ay1)v=0.

Since we know that y1(x) is a solution to the original problem, the coefficient of the last term is equal to zero. Furthermore, substituting y1(x) into the second term's coefficient yields (for that coefficient) 2a(−b2ae−b2ax)+be−b2ax=(−b+b)e−b2ax=0.

Therefore, we are left with ay1v″=0.

Since a is assumed non-zero and y1(x) is an exponential function (and thus always non-zero), we have v″=0.

This can be integrated twice to yield v(x)=c1x+c2 where c1,c2 are constants of integration. We now can write our second solution as y2(x)=(c1x+c2)y1(x)=c1xy1(x)+c2y1(x).

Since the second term in y2(x) is a scalar multiple of the first solution (and thus linearly dependent) we can drop that term, yielding a final solution of y2(x)=xy1(x)=xe−b2ax.

Finally, we can prove that the second solution y2(x) found via this method is linearly independent of the first solution by calculating the Wronskian W(y1,y2)(x)=|y1xy1y1′y1+xy1′|=y1(y1+xy1′)−xy1y1′=y12+xy1y1′−xy1y1′=y12=e−bax≠0.

Thus y2(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 y1(t) of the homogeneous equation [r(t)=0], let us try a solution of the full non-homogeneous equation in the form: y2=v(t)y1(t) where v(t) is an arbitrary function. Thus y2′=v′(t)y1(t)+v(t)y1′(t) and y2″=v″(t)y1(t)+2v′(t)y1′(t)+v(t)y1″(t).

If these are substituted for y, y′, and y″ in the differential equation, then y1(t)v″+(2y1′(t)+p(t)y1(t))v′+(y1″(t)+p(t)y1′(t)+q(t)y1(t))v=r(t).

Since y1(t) is a solution of the original homogeneous differential equation, y1″(t)+p(t)y1′(t)+q(t)y1(t)=0, so we can reduce to y1(t)v″+(2y1′(t)+p(t)y1(t))v′=r(t) which is a first-order differential equation for v′(t) (reduction of order). Divide by y1(t), obtaining v″+(2y1′(t)y1(t)+p(t))v′=r(t)y1(t).

One integrating factor is given by μ(t)=e∫(2y1′(t)y1(t)+p(t))dt, and because

∫(2y1′(t)y1(t)+p(t))dt=2∫y1′(t)y1(t)dt+∫p(t)dt=2ln⁡(y1(t))+∫p(t)dt=ln⁡(y12(t))+∫p(t)dt,

this integrating factor can be more neatly expressed as μ(t)=eln⁡(y12(t))+∫p(t)dt=y12(t)e∫p(t)dt. Multiplying the differential equation by the integrating factor μ(t), the equation for v(t) can be reduced to ddt(v′(t)y12(t)e∫p(t)dt)=y1(t)r(t)e∫p(t)dt.

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: y2(t)=v(t)y1(t).

## See also

* Variation of parameters

## References

* Boyce, William E.; DiPrima, Richard C. (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). Hoboken, NJ: John Wiley & Sons, Inc.. ISBN 978-0-471-43338-5.
* Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. 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.

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