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*) + C*cg*(*x*), where:

*f*′ + λ*f*=*p*(*x*),*g*′ + λ*g*= 0,- C = the constant of integration.

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

*g*= C

*e*

^{-λx}

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

**The solution is y = f(x) + Ce ^{-λ(x)}**

**Note**: A solution of the homogeneous equation has a form 𝑥(𝑡) = 𝑒^{𝑚𝑡}.

## 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 (m_{1}, m_{2}) |
x = Ae^{m1t} + Be^{m2t} |

One real root m, repeated | x = (A + Bt)e^{mt} |

Complex conjugate roots α ± iΒ | x = e^{αt}(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