Calculus How To

Complementary Function

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A complementary function is one part of the solution to a linear, autonomous differential equation.

An ordinary differential equation (ODE) relates the sum of a function and its derivatives. When the explicit functions y = f(x) + cg(x) form the solution of an ODE, g is called the complementary function; f is the particular integral.

Example of Solution Using a Complementary Function

Example question: Solve the following differential equation, using a complementary function and a particular integral:
y′ + λ y = p(x).

In order to solve this, you have an important piece of information. From the definition above, you’re looking for a solution with the form y = f(x) + Ccg(x), where:

Step 1: Solve g′ + λ g = 0 to find g. By separating the variables, this becomes:
g = Cex


Step 2: Plug the value you found in Step 1 into the general formula given above: y = f(x) + Ccg(x):

The solution is y = f(x) + Ce-λ(x)

Note: A solution of the homogeneous equation has a form 𝑥(𝑡) = 𝑒𝑚𝑡.

Complementary and Auxiliary Equations

When solving differential equations, it’s sometimes necessary to solve the auxiliary equation (a quadratic equation) before you can find the complementary function. Different forms of the auxiliary equation will lead to different forms of the complementary function:

Form of auxiliary equation Resulting complementary function
two real roots (m1, m2) x = Aem1t + Bem2t
One real root m, repeated x = (A + Bt)emt
Complex conjugate roots α ± iΒ x = eαt(A cos (Βt) + B sin (Βt))

References

Jeffrey, A. (2004). Mathematics for Engineers and Scientists 6th Edition. Chapman and Hall/CRC.
Scroggs, M. 4.3 Complementary functions and particular integrals
Solving ODEs by using the Complementary Function and Particular Integral. Retrieved December 26, 2019 from: http://www.mathematics.me.uk/tutorials/Solving%20ODEs%20by%20Complementary%20Function%20and%20%20Particular%20Integral.pdf


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