A central goal in differential equations is learning how to find their solutions. In fact, most problems you’ll encounter begin with a familiar prompt:
In algebra, a solution is any value that “satisfies” the algebraic equation. In contrast, a solution to a differential equation is any function that “satisfies” the differential equation. Understanding what it means for a function to satisfy a differential equation is the first step toward understanding its solutions.
In differential equations, a solution is a function that satisfies the equation. This means that when you plug it into the dependent variable, both sides simplify to the same expression.
To verify a solution, take any necessary derivatives, substitute into the equation, and simplify. If both sides match, or reduce to something like \(0 = 0\text{,}\) then the function is a solution.
