Auxiliary Equation

Second Order Differential Equation

The auxiliary equation of a second order differential equation is defined as

$$a{r}^2+br+c=0$$

Three Cases of the Differential Equation

Three cases to the nature of the equation are to be considered and depend all on the discriminant $b^2-4ac$

$b^2-4ac>0$

In this case the equation has two real and unequal roots $r\_1$ and $r\_2$. If $r\_1$ and $r\_2$ are two real and unequal roots of the auxiliary equation $a{r}^2+br+c=0$ then $$ y = c_1e^{r_1x}+ c_2e^{r_2x}

is the general solution to $ay''+by'+cy = 0$ and

y_1 = e^{rx}

y_2 = xe^{rx}

##### $b^2-4ac = 0$ In this case $r\_1 = r\_2 = -b/2a$. When only one real root $r$ to the auxiliary equation exists the general solution to the equation is

y = c_1e^{rx}+c_2xe^{rx}

##### $b^2 - 4ac < 0$ Here the auxiliary equation will yield two complex roots $r\_1 = \alpha + i\beta$ and $r\_2 = \alpha - i\beta$, where $\alpha, \beta$ are real. Thus two complex linearly independent solutions arise

y_1 = e^{(\alpha+i\beta)x} = e^{\alpha x}(cos\beta x + i\ sin\beta x)

$$and$$

y_2 = e^{(\alpha-i\beta)x} = e^{\alpha x}(cos\beta x - i\ sin\beta x)

From $y\_1$ and $y\_2$ we can obtain two _real-valued_ solutions. Their derivation is explained clearly in [[Extracting the Fundamental System out of Complex Conjugate Pairs|extracting the fundamental real system out of complex conjugate pairs]]. These equations are:

y_3 = \frac{1}{2}y_1 + \frac{1}{2}y_2 = e^{\alpha x}cos\beta x

$$and$$

y_4 = \frac{1}{2i}y_1 - \frac{1}{2i}y_2 = e^{\alpha x}sin\beta x

such that the general solution to $ay'' + by' + cy = 0$ is

y = e^{\alpha x}(c_1\ cos\beta x + c_2\ sin \beta x)