2.2.1.1 Solved using first_order_ode_autonomous

Entering first order ode autonomous solver

\[\begin {aligned} x^{\prime }&=-\lambda x \end {aligned}\]

Integrating gives

\begin{align*} \int -\frac {1}{\lambda x}d x &= dt\\ -\frac {\ln \left (x \right )}{\lambda }&= t +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} -\lambda x&= 0 \end{align*}

for \(x\). This is because of dividing by the above earlier. This gives the following singular solution(s), which also has to satisfy the given ODE.

\begin{align*} x&=0 \end{align*}

Solving for \(x\) from \(-\frac {\ln \left (x\right )}{\lambda } = t +c_1\) gives

\begin{align*} x &= {\mathrm e}^{-c_1 \lambda -t \lambda } \\ \end{align*}

Solutions plot

Summary of solutions found

\begin{align*} x&={\mathrm e}^{-c_1 \lambda -t \lambda }\\ x&=0 \end{align*}