\begin {align*} y^{\prime } & =f\left ( x\right ) \\ y^{\prime } & =f\left ( y\right ) \end {align*}
The following flow chart gives the algorithm for solving quadrature ode.
ode internal name "quadrature"
Solved by direct integration. There are two forms. They are
\begin {align*} y^{\prime } & =f\left ( x\right ) \\ y^{\prime } & =f\left ( y\right ) \end {align*}
For first form, the solution is \[ y=\int f\left ( x\right ) dx+c \] For the second form the solution is \begin {align*} \int \frac {dy}{f\left ( y\right ) } & =\int dx\qquad f\left ( y\right ) \neq 0\\ \int \frac {dy}{f\left ( y\right ) } & =x+c \end {align*}
For the second form, we have to consider singular solutions also, these are found by solving for \(y\) from \(f\left ( y\right ) =0\) and to see if these solutions can also be obtained from the general solution for any value of \(c\). These two forms are special cases of separable first order ode \(y^{\prime }=f\left ( x\right ) g\left ( y\right ) .\)