Complex Roots, Repeated Roots, and Reduction of Order

The characteristic equation $a{r}^2+br+c=0$ has three outcomes, fixed by the sign of the discriminant $b^2-4ac$. Two distinct real roots ($b^2-4ac>0$) give $y=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}$. The remaining two cases break that form. With complex conjugate roots ($b^2-4ac<0$), Euler's formula converts the complex exponentials into real-valued solutions. With a repeated real root ($b^2-4ac=0$), only one exponential solution exists, and reduction of order supplies the second by a method that applies far beyond constant coefficients.

Complex roots and Euler's formula

Suppose $b^2-4ac<0$. The quadratic formula gives conjugate roots

with $\lambda ,\mu$ real and $\mu >0$. Formally $y_1=e^{(\lambda +i\mu )t}$ and $y_2=e^{(\lambda -i\mu )t}$ are solutions, but they are complex-valued, whereas the equation has real coefficients and we want real solutions. Euler's formula defines the complex exponential through real functions.

The formula is obtained by substituting $i\theta$ into the Taylor series for the exponential and separating the powers of $i$: the even powers reproduce the cosine series and the odd powers, carrying one factor of $i$, reproduce the sine series. Adopting it as the definition of the complex exponential makes the usual law $e^{(\lambda +i\mu )t}=e^{\lambda t}{e}^{i\mu t}$ hold and gives a concrete complex number with real and imaginary parts built from elementary real functions.

To extract real solutions, use a structural fact about the operator: because $p$ and $q$ are real, the real and imaginary parts of a complex-valued solution are each real-valued solutions on their own.

Substituting $u+iv$ into $L$ and using that $p,q$ are real splits $L[u+iv]=L[u]+iL[v]$. A complex number is zero exactly when its real and imaginary parts both vanish, so $L[y]=0$ forces $L[u]=0$ and $L[v]=0$. Applying this to $y_1=e^{\lambda t}\cos (\mu t)+i{e}^{\lambda t}\sin (\mu t)$ yields the real pair

Their Wronskian is $W[u,v](t)=\mu e^{2\lambda t}$, which is nonzero since $\mu \ne 0$, so $u,v$ form a fundamental set and the general solution is

The sign of $\lambda$ controls the envelope and $\mu$ controls the oscillation rate. When $\lambda <0$ the amplitude shrinks (a decaying oscillation); when $\lambda >0$ it grows; when $\lambda =0$ the roots are purely imaginary $\pm i\mu$ and the solution is a pure sinusoid $c_1\cos (\mu t)+c_2\sin (\mu t)$ of constant amplitude. The three regimes are the underdamped, unstable, and undamped cases of a mechanical or electrical oscillator.

The repeated root

Now $b^2-4ac=0$, so the quadratic has a double root

Only one exponential $y_1=e^{-bt/(2a)}$ emerges, and a fundamental set needs a second solution not proportional to it. Reduction of order, due to d'Alembert, produces it: since any constant multiple $c{y}_1$ is a solution, replace the constant by a function and look for a solution $y=v(t)y_1(t)$.

The new solution is $y_2=t{e}^{-2t}$: the same exponential multiplied by $t$. Its Wronskian with $y_1$ is $W[e^{-2t},t{e}^{-2t}]=e^{-4t}\ne 0$, so $\{e^{rt},t{e}^{rt}\}$ is a fundamental set and the general solution for a repeated root $r$ is

The cancellation is general. When $r$ is a double root, both $a{r}^2+br+c$ and its derivative $2ar+b$ vanish at $r$; since these are the very coefficients of $v$ and $v^{\prime }$, both terms drop out and leave $v^{\prime \prime }=0$.

Reduction of order in general

The substitution $y=v(t)y_1(t)$ is not tied to constant coefficients. Given any single solution $y_1$ of the variable-coefficient equation

it produces a second, independent solution. Substituting $y=v{y}_1$ and using that $y_1$ itself solves the equation, the $v$-only terms vanish and what remains is

There is no undifferentiated $v$: the equation is first-order in $w=v^{\prime }$. It separates,

and one integration of $w$ recovers $v$, hence $y_2=v{y}_1$. The order of the equation has been reduced from two to one, which is where the method's name comes from and why it works even when no characteristic equation is available.

The repeated-root form $t{e}^{rt}$ is the constant-coefficient special case: with $y_1=e^{rt}$ and $p=-2r$, the integral gives $w=$ const and $v=t$.

The three cases side by side

Every row is a two-dimensional solution space with a nonzero Wronskian; the cases differ only in how the second basis function is built.¹²³