3.8.3—An initial condition to find a

An initial condition such as y(x₀)=y₀ is substituted into the integrated family to determine its constant. For a second-order equation, conditions on y and y′ may be needed.

Apply the condition after integrating, not before; if the problem gives a physical initial velocity, translate it into y′(x₀) explicitly.

If y′=3x²−2 and y(1)=4, then y=x³−2x+C and C=5.

The initial point is not automatically x=0; use the coordinate stated in the question.