Euler's formula for solutions
/ OY-ler /
When the characteristic equation throws up complex roots, you are momentarily handed a solution like e^((1 + 2i)x) — a perfectly good function, but one written with an imaginary exponent, which is hard to picture and certainly not the real-valued answer a physical problem demands. Euler's formula is the bridge that turns these imaginary exponentials into honest cosines and sines.
The formula is e^(i theta) = cos(theta) + i sin(theta). It says that raising e to an imaginary power produces a point on the unit circle: the real part is the cosine, the imaginary part is the sine. For ODEs we use it on the conjugate roots alpha ± beta i. Write e^((alpha + beta i)x) = e^(alpha x) e^(i beta x) = e^(alpha x)(cos(beta x) + i sin(beta x)). Both the real part e^(alpha x) cos(beta x) and the imaginary part e^(alpha x) sin(beta x) are themselves real solutions of the (real) equation — a fact guaranteed because the coefficients are real, so the real and imaginary parts of a complex solution must separately solve the equation. Those two real functions are the answer we actually report.
Euler's formula is the secret reason that exponentials and oscillations are the same thing in disguise. A decaying spring (cos and sin under a decaying envelope) and a decaying exponential are both special cases of e^(r x) for different r — real r gives decay, imaginary r gives oscillation, complex r gives both at once. This unification is why the single exponential guess y = e^(r x) is enough to solve every constant-coefficient linear equation: the formula quietly converts the complex cases back into the trigonometric functions you can plot.
From the root -1 + 2i: e^((-1+2i)x) = e^(-x)(cos 2x + i sin 2x). The real part e^(-x) cos 2x and imaginary part e^(-x) sin 2x are the two real solutions. Setting theta = pi in the formula also yields the famous e^(i pi) + 1 = 0.
Euler's formula splits a complex exponential solution into its real cosine and sine parts.
This split into real and imaginary solutions works only because the ODE has real coefficients. With complex coefficients the real and imaginary parts of a solution need not separately solve the equation, and the cos/sin reduction can fail.