Verifying a Solution to dy/dx = 2x

Differential Equations · Axiom Academy

EXAMPLE Verifying a Solution to Confirm that y = x^2 + C satisfies the differential equation by differentiating and comparing both sides. Show that y = x^2 + C is a solution of the differential equation , where C is an arbitrary constant. To verify a proposed solution, we differentiate it and check whether the result matches the right-hand side of the differential equation exactly. Nice work — you've verified a solution to a differential equation by direct substitution. Here's what carries forward: Verification process: substitute the proposed solution into the differential equation and confirm both sides match exactly — no guessing, no solving from scratch. Differentiation is the tool: differentiate the candidate solution and compare the result to the equation's right-hand side. The constant C : for any real number C , so C always drops out of the check. That's why y = x^2 + C works for every choice of C — it's a whole family of solutions, not just one curve. General vs. particular solutions: y = x^2 + C is the general solution (every member of the family). A particular solution fixes one value of C , usually from an initial condition. Geometric picture: different values of C give vertically shifted copies of y = x^2 . A vertical shift never changes the slope at a given x , so every shifted parabola still satisfies .

This is the written version of the interactive lesson above. See the full Differential Equations course.