Verifying Solutions

Calculus 2 · Axiom Academy

Confirm a proposed function solves a differential equation and its initial condition. Verify that y = e^ x^2 is a solution of the initial value problem below. A complete check has two parts: the function must satisfy the differential equation, and it must satisfy the initial condition. Nicely done. You verified a differential equation solution end to end. Here is what carried the argument: Verification is two checks: a complete solution must satisfy the differential equation and every initial condition — passing one alone is not enough. Differentiate first: find from the proposed y , watching the chain rule on composite functions like e^ x^2 . Substitute, then compare: plug y and into the equation and confirm the two sides reduce to the same expression — here both became . Test the initial condition directly: substituting x=0 gives y(0)=e^ 0 =1 , matching the required value. Verifying is easier than solving: you never had to solve the equation — checking a candidate only takes differentiation and substitution. Both conditions hold, so y = e^ x^2 is the complete solution of the initial value problem.

This is the written version of the interactive lesson above. See the full Calculus 2 course.