second order linear constant coeﬀ

2.2.7.1 second order linear constant coeﬀ

\[\begin {aligned} x^{\prime \prime }+3 x^{\prime }&=0 \end {aligned}\]

Entering second order linear constant coeﬃcient ode solverThis is second order with constant coeﬃcients homogeneous ODE. In standard form the ODE is

\[ A x''(t) + B x'(t) + C x(t) = 0 \]

Where in the above \(A=1, B=3, C=0\). Let the solution be \(x=e^{\lambda t}\). Substituting this into the ODE gives

\[ \lambda ^{2} {\mathrm e}^{\lambda t}+3 \lambda \,{\mathrm e}^{\lambda t} = 0 \tag {1} \]

Since exponential function is never zero, then dividing Eq(2) throughout by \(e^{\lambda t}\) gives

\[ \lambda ^{2}+3 \lambda = 0 \tag {2} \]

Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form. Using the quadratic formula the roots are

\[ \lambda _{1,2} = \frac {-B}{2 A} \pm \frac {1}{2 A} \sqrt {B^2 - 4 A C} \]

Substituting \(A=1, B=3, C=0\) into the above gives

\begin{align*} \lambda _{1,2} &= \frac {-3}{(2) \left (1\right )} \pm \frac {1}{(2) \left (1\right )} \sqrt {3^2 - (4) \left (1\right )\left (0\right )}\\ &= -{\frac {3}{2}} \pm {\frac {3}{2}} \end{align*}

Hence

\begin{gather*} \begin {aligned} \lambda _1 &= -{\frac {3}{2}} + {\frac {3}{2}}\\ \lambda _2 &= -{\frac {3}{2}} - {\frac {3}{2}} \end {aligned} \end{gather*}

Which simpliﬁes to

\begin{gather*} \begin {aligned} \lambda _1 &= 0\\ \lambda _2 &= -3 \end {aligned} \end{gather*}

Since the roots are distinct, the solution is

\begin{gather*} \begin {aligned} x &= c_1 e^{\lambda _1 t} + c_2 e^{\lambda _2 t}\\ x &= c_1 e^{\left (0\right )t} +c_2 e^{\left (-3\right )t} \end {aligned} \end{gather*}

\begin{align*} x &=c_1 +c_2 \,{\mathrm e}^{-3 t} \end{align*}

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