Here is a way to find the particular solution of a separable differential equation without using the +*C *and it might even be a little faster. As an example consider the initial value problem

Separate the variables as usual and then write each side a definite integral with the lower limit of integration the number from the initial condition and the upper limit variable. I’ll replace the dummy variable in the integrand with an upper-case *X* or *Y* to avoid having *x* and *y* in two places.

Integrate

Note that near the initial condition where *y* = 0, so . Continue and solve for *y.*

No muss; no fuss.

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