Owl: Some differential equations let you rearrange so every y huddles with dy and every x with dx: g(y) dy = f(x) dx. Integrate both sides, solve for y when you can, then apply initial conditions.
Can you split dy/dx = xy into "y stuff" and "x stuff"? Yes — carefully. Division can lose constant solutions like y = 0, and domains matter after logarithms appear.
We'll separate, integrate, and pin down C — without throwing away solutions we divided by.