First-order separable ODEs
dy/dx = f(x)g(y). Rearrange: dy/g(y) = f(x) dx. Integrate both sides.
Separable ODEs are the simplest. Identify f(x) and g(y); separate variables on each side; integrate; rearrange for y if possible. Constant of integration combines into a single C.
Worked examples
dy/dx = xy: dy/y = x dx ⇒ ln|y| = x²/2 + C ⇒ y = Aex²/2.
dy/dx = (1+y)/(1+x): ln|1+y| = ln|1+x| + C ⇒ 1+y = A(1+x).
Initial value y(0) = 2 fixes A.
Frequently asked questions
Always works?
Only when ODE is separable in this form. Check first.
Why one constant?
‘A constant + a constant = a constant’. We collapse them into one final C or A.