08 Constant Coefficient
2nd 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 \(c_2\)