Initial Value Problems

Calculus 2 · Axiom Academy

How a single initial condition selects one unique solution from a whole family of curves. 1. A Differential Equation Gives a Family Integrating undoes the derivative, but the antiderivative is only fixed up to a constant. So the general solution is an entire family of parabolas — identical in shape, just slid up or down by C : One equation → infinitely many curves, one per value of C An initial value problem pairs the differential equation with a point the solution must pass through. Suppose we require the curve through (1, 3) : The condition y(1) = 3 reads: "when x = 1 , y must equal 3 ." Substitute that point into the family y = x^2 + C and solve for the one constant that fits: The unique solution: y = x^2 + 2 sets the shape shared by every curve in the family. y(1) = 3 sets the position — it selects C = 2 and discards the rest. Why does one point give exactly one curve — never zero, never two? Slide the target height up the line x = 1 : for each height there is one and only one member of the family threading it, because y(1) = 1 + C ties each height to a single constant. Pick a height, pick a curve. This is no accident of our example. The Existence and Uniqueness Theorem guarantees it for any well-behaved equation: A solution through the point is guaranteed to exist — the curve is always there to find. There is only one such curve — two solution curves of a nice equation never cross the same point.

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