Reduction of order explained

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

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$$a\; \backslash left(v$$ 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$$\backslash left(a\; y\_1\; \backslash right)\; v$$ + \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)$$2\; a\; \backslash left(-\; \backslash frac\; e^\; \backslash right)\; +\; b\; e^\; =\; \backslash left(-b\; +\; b\; \backslash right)\; e^\; =\; 0.$$

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

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</math>], 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$$y\_2\text{'}\; =\; v\text{'}(t)y\_1(t)\; +\; v(t)y\_1\text{'}(t)$$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$$y\_1(t)\backslash ,v$$ + (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$$y\_1(t)\backslash ,v$$ + (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.

One integrating factor is given by

\mu(t) = e ^{\int( (2 y _{1'(t)})/(+ p(t))dt y _{1(t)})} , and because\int \left( (2 y _{1'(t)})/(y _{1(t)}) + p(t)\right)dt = 2 \int (y _{1'(t)})/(y _{1(t)}) dt + \int p(t)dt = 2 ln (y _{1(t))} + \int p(t)dt = ln (y ^{2 (t))} _{1} + \int p(t)dt, this integrating factor can be more neatly expressed as \mu(t) = e ^{ln (y ^{2 (t))} _{1} + \int p(t)dt} = y ^{2 (t)e} _{1} ^{\int} . Multiplying the differential equation by the integrating factor \mu(t) , the equation for v(t) can be reduced to$$\backslash frac\backslash left(v\text{'}(t)\; y\_1^2(t)\; e^\backslash right)\; =\; y\_1(t)r(t)e^.$$

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

• Book: Boyce . William E. . DiPrima . Richard C. . Elementary Differential Equations and Boundary Value Problems . 2005 . John Wiley & Sons, Inc. . 978-0-471-43338-5 . 8th . Hoboken, NJ.
• Book: Teschl . Gerald Teschl

. Gerald . Gerald Teschl . Ordinary Differential Equations and Dynamical Systems . . . 2012 . 978-0-8218-8328-0 .

• Eric W. Weisstein, Second-Order Ordinary Differential Equation Second Solution, From MathWorld—A Wolfram Web Resource.