2.1.2.1 Solved using first_order_ode_autonomous

Entering first order ode autonomous solver

\[\begin {aligned} y^{\prime }+c y&=a \end {aligned}\]

Integrating gives

\begin{align*} \int \frac {1}{-c y +a}d y &= dx\\ -\frac {\ln \left (-c y +a \right )}{c}&= x +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} -c y +a&= 0 \end{align*}

for \(y\). 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*} y&=\frac {a}{c} \end{align*}

Solving for \(y\) from \(-\frac {\ln \left (-c y+a \right )}{c} = x +c_1\) gives

\begin{align*} y &= -\frac {{\mathrm e}^{-c_1 c -x c}-a}{c} \\ \end{align*}

Summary of solutions found

\begin{align*} y&=-\frac {{\mathrm e}^{-c_1 c -x c}-a}{c}\\ y&=\frac {a}{c} \end{align*}