Electricity and Magnetism · Problem 8.2

MAXWELL'S EQUATIONS AND ELECTROMAGNETIC WAVES — Edward Purcell & David Morin

Physics — E&M · solving-linear-differential-equations

Problem

Consider the nnth-order homogeneous linear differential equation andnxdtn+an−1dn−1xdtn−1+⋯+a1dxdt+a0x=0.a_n \frac{d^n x}{dt^n} + a_{n-1}\frac{d^{n-1}x}{dt^{n-1}} + \cdots + a_1 \frac{dx}{dt} + a_0 x = 0. Show that the solutions take the form of x(t)=Aieritx(t) = A_i e^{r_i t}, where the rir_i depend on the aja_j coefficients. Hint: If the (d/dt)(d/dt) derivatives were replaced by the letter zz, then we would have an nnth-order polynomial in zz, which we know can be factored, by the fundamental theorem of algebra. (You can assume that the roots of this polynomial are distinct. Things are a little more complicated if there are double roots; this is discussed in the solution.)

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