Differential Equations › Section C · Second order differential equations › Topic C.3
Complex roots
When the characteristic equation has no real roots, the solutions oscillate. Euler's formula turns a pair of complex exponentials into real sines and cosines, and the real part of the root decides whether the oscillation fades, grows or holds steady.
In this topic
- When the discriminant is negative
- Euler's formula
- From complex to real solutions
- Using the formula
- Worked example: a growing oscillation
- Worked example: conditions away from zero
- Reading the solution
- Common mistakes to avoid
- Key takeaways
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