08 Constant Coefficient

Last Updated: 2 years ago2023-01-25 ; Contributors: AhmedThahir

2^nd Order Homogeneous DE with constant coefficients

$$y'' + py' + qy = 0$$

where $p, q$ are constants

Consider $y = e^{mx}$ as a possible solution, where $m$ = unknown constant. So our goal is to find $m$.

Then

$$y' = m \cdot e^{mx} \\ y' = m^2 \cdot e^{mx} \\ \implies (m^2 \cdot e^{mx}) + p(m \cdot e^{mx}) + qe^{mx} = 0 \\ e^{mx} ( m^2 + pm + q ) = 0 \\$$

Auxiliary equation

$$e^{mx} \ne 0 \\ \implies ( m^2 + pm + q ) = 0$$

Solve this to get the value(s) of unknown $m$

Roots		General Solution $y$
real and distinct	$m_1, m_2$	$c_1 e^{m_1 x} + c_2 e^{m_2 x}$
equal roots	$m_1 = m_2 = m$	$e^{mx} (c_1 + c_2 x )$
Complex roots	$m_1, m_2 = a \pm ib$	$e^{ax} (c_1\cos bx + c_2 \sin bx )$

Boundary Value Problems

Using given ‘initial conditions’, we need to find the values of $c_1$ and $c_2$

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