Recipe

Solve a separable differential equation

Use when: dy/dx factors as a function of x alone times a function of y alone

Goal: the general solution, a family of curves with one arbitrary constant, ready to be pinned down by an initial condition

Taught in Separable equations · a later section extends this recipe with more cases

1. 1

   Try to factor the right side into a piece depending only on x times a piece depending only on y.

   Example

   \dfrac{dy}{dx}=x^2y: the right side is already x^2\cdot y, a function of x alone times a function of y alone.

   Why: this is an algebra check, not a calculus step; an equation like \dfrac{dy}{dx}=x+y can't be factored this way no matter how it's rearranged, and separation simply doesn't apply to it.

2. 2

   Move every y (and dy) to one side and every x (and dx) to the other.

   Example

   \frac{dy}{y} = x^2\,dx
   • h(y)=0 for some constant y=c — that constant function is also a solution, set aside by the division in this step; report it separately alongside the family below.
   • h(y)\neq0 on the range of interest — the division is valid and the general family below covers every solution.

   Why: dividing by h(y) requires h(y)\neq0, and any root of h gives up a constant solution the division step discards.

3. 3

   Integrate the y-side with respect to y and the x-side with respect to x, combining both constants of integration into one.

   Example

   \int\frac{dy}{y} = \int x^2\,dx \implies \ln|y| = \frac{x^3}{3}+C_1

   Why: each side is now an ordinary single-variable integral, and the two arbitrary constants merge into one because their difference is itself just another arbitrary constant.

4. 4

   Exponentiate, or otherwise isolate y, when doing so doesn't force an unnecessary split into multiple branches. Leave the solution implicit when it would.

   Example

   Exponentiating: |y|=e^{x^3/3+C_1}=e^{C_1}e^{x^3/3}. Absorbing the sign and e^{C_1} into a single renamed constant C (allowed to be any nonzero real, or 0 from the constant solution): y = Ce^{x^3/3}

   Why: forcing an explicit solution when it would split into a plus-or-minus or multiple implicit branches loses no information but adds unnecessary case-splitting the implicit form avoids.

5. 5

   Report the solution with its arbitrary constant intact; every value of C traces one member of the family.

   Example

   \boxed{y = Ce^{x^3/3}}
   • Not yet: a specific starting value y(x_0)=y_0 is given, pinning C down to one number — covered in Initial value problems and particular solutions

   Why: without an initial condition, every value of C gives an equally valid solution, so the family, not a single curve, is the complete answer at this stage.