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Second-Order Ordinary Differential Equation Second Solution


If one solution (y_1) to a second-order ordinary differential equation

 y^('')+P(x)y^'+Q(x)y=0
(1)

is known, the other (y_2) may be found using the so-called reduction of order method. From Abel's differential equation identity

 (dW)/W=-P(x)dx,
(2)

where

 W=y_1y_2^'-y_1^'y_2
(3)

is the Wronskian.

Integrating gives

 int_a^x(dW)/W=-int_a^xP(x^')dx^'
(4)
 ln[(W(x))/(W(a))]=-int_a^xP(x^')dx^',
(5)

and solving for W(x) gives

 W(x)=W(a)exp[-int_a^xP(x^')dx^'].
(6)

But

 W=y_1y_2^'-y_1^'y_2=y_1^2d/(dx)((y_2)/(y_1)),
(7)

so combining (◇) and (◇) yields

 d/(dx)((y_2)/(y_1))=W(a)(exp[-int_a^xP(x^')dx^'])/(y_1^2)
(8)
 y_2(x)=y_1(x)W(a)int_b^x(exp[-int_a^(x^')P(x^(''))dx^('')])/([y_1(x^')]^2)dx^'.
(9)

Disregarding W(a), since it is simply a multiplicative constant, and the constants a and b, which will contribute a solution which is not linearly independent of y_1, leaves

 y_2(x)=y_1(x)int^x(exp[-int^(x^')P(x^(''))dx^('')])/([y_1(x^')]^2)dx^'.
(10)

In the special case P(x)=0, this simplifies to

 y_2(x)=y_1(x)int^x(dx^')/([y_1(x^')]^2).
(11)

If both general solutions to a second-order nonhomogeneous differential equation are known, variation of parameters can be used to find the particular solution.


See also

Abel's Differential Equation Identity, Second-Order Ordinary Differential Equation, Undetermined Coefficients Method, Variation of Parameters

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Cite this as:

Weisstein, Eric W. "Second-Order Ordinary Differential Equation Second Solution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Second-OrderOrdinaryDifferentialEquationSecondSolution.html

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