Find two linearly independent solutions of the 2nd-order linear homogeneous differential equation
$$y'' + 1y' - 6y = 0$$
The linearly independent solutions are
$$y_1 = $$
$$y_2 = $$
and the general solution of the equation is
$$y(x) = $$
where $c_1$, and $c_2$ are arbitrary constants.
(Note: Let $m_1$ and $m_2$ be two solutions of the auxiliary (characteristic) equation.
(i) If $m_1 < m_2$, write $y_1 = e^{m_1x}$ and $y_2 = e^{m_2x}$;
(ii) If $m_1 = m_2$, write $y_1 = e^{m_1x}$ and $y_2 = xe^{m_1x}$; and
(iii) If $m_1 = a + bi$, and $m_2 = a - bi$, write $y_1 = e^{ax}cos(bx)$ and $y_2 = e^{ax}sin(bx)$.)
