Used to solve [[Ordinary Differential Equations (ODEs)|ODEs]] We can use the general form of a first order homogeneous ODE to inspire a solution for the $n$-th order version. $ \alpha_1\frac{dy}{dt} +\alpha_0y=0$ The general solution to this is: $ y = Ce^{-\frac{\alpha_0}{\alpha_1}t} $ If we assume $y=Ce^{\lambda t}$ and plug this into the $n$-th order equation, we get: $ \alpha_n\frac{d^n}{dt^n}Ce^{\lambda t}+\alpha_{n-1}\frac{d^{n-1}}{dt^{n-1}}Ce^{\lambda t}+\dots+\alpha_1\frac{d}{dt}Ce^{\lambda t}+\alpha_0Ce^{\lambda t}=0 $ Simplifying derivatives: $ \alpha_n\lambda^nCe^{\lambda t}+\alpha_{n-1}\lambda^{n-1}Ce^{\lambda t}+\dots+\alpha_1\lambda Ce^{\lambda t}+\alpha_0Ce^{\lambda t}=0 $ Finally, canceling out $Ce^{\lambda t}$ yields: $ \alpha_n\lambda^n+\alpha_{n-1}\lambda^{n-1}+\dots\alpha_1\lambda+\alpha_0=0 $ This is known as the characteristic equation/polynomial of the ODE. Finding the roots of this equation provides viable values for $\lambda$ in our solution. ## Repeated Roots - [[Reduction of Order]] ## Complex Roots Consider the following second order, constant coefficient ODE: $ y''+y=0,\quad y(0)=\alpha,\quad y'(0)=\beta $ Substituting the exponential guess $y=Ce^{\lambda t}$ results in the characteristic polynomial $ \lambda^2+1=0 $ which has complex conjugate roots $\lambda=\pm i$ Our general solution becomes a copmlex function: $ y=K_1e^{it}+K_ 2e^{-it} $ with $K_1$ and $K_2$ being potentially complex coefficients. To split the exponentials into real and imaginary parts, the power series representation is used: $ e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\dots $ Substituting $x=it$: $ \begin{align} e^{it}&=1+it-\frac{t^2}{2}+\frac{t^3}{3!}+\dots\\ &=\left(1-\frac{t^2}{2}+\frac{t^4}{4!}-\frac{t^6}{6!}-\dots\right)+i\left(t-\frac{t^3}{3!}+\frac{t^5}{5!}-\dots\right) \end{align} $ This results in the Euler identity: $ e^{it}=\cos(t)+i\sin(t) $ and similarly: $ e^{-it}=\cos(t)-i\sin(t) $ Selecting $K_1,\,K_2=\frac{1}{2}$, we find: $ y_1=\cos(t) $ While selecting $K_1=\frac{i}{2}$ and $K_2=-\frac{i}{2}$ results in: $ y_2=\sin(t) $ We can now generalize that for a second order, linear, constant coefficient, homogeneous ODE with complex roots $\lambda=\mu\pm i\sigma$: $ \boxed{y=C_1e^{\mu t}\cos(\sigma t)+C_2e^{\mu t}\sin(\sigma t)} $