What is a Differential Equation?

Calculus 2 · Axiom Academy

LESSON What is a Differential Equation? Run the derivative backwards: instead of solving for a number, you solve for a whole function whose rate of change obeys a rule. An algebraic equation like x^2 = 9 asks " which number works?" A differential equation asks " which function works?" Take : we need a function y(x) whose slope at every point is 2x . Watch the solution y = x^2 — its tangent line tilts to match 2x at every x . 2. Order: the Highest Derivative The order of a differential equation is simply the highest derivative that appears in it. is first-order ; y'' + y = 0 is second-order . Below, solves y'' + y = 0 : at every instant the second derivative is the exact mirror of , so their sum stays zero. Highest derivative is y' . Example: . Highest derivative is y'' . Example: y'' + y = 0 . 3. Checking a Solution by Substituting You do not have to solve a differential equation to know whether a given function is a solution — you can verify it. Differentiate the candidate, substitute, and see if both sides agree. Test y = x^3 + 5 against : its actual slope (left) tracks the required 3x^2 (right) and they match at every x . Take a candidate function y(x) . Compute the derivatives it needs. Check that both sides are equal. The constant shifts the curve up or down but does not change its slope, so regardless of the constant. That freedom is exactly what the next step is about. 4. General vs. Particular Solutions

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