We evaluate the left-hand side of the equation at *x* = 4: (4)^{2} + 6 = 22. Then we evaluate the right-hand side of the equation at *x* = 4: 4(4) + 11 = 27. Since we don't get the same result from both sides of the equation, *x* = 4 is not a solution to the equation. In differential equations the variables stand for functions instead of numbers. In the differential equation *y*' = *xy*,
the symbols *y* and *y*' stand for functions. Therefore a solution to a differential equation is a function rather than a number. As a Twinkie satisfies a sweet tooth, a **solution** to a differential equation is a function that *satisfies* that d.e. (in other words, makes the d.e. true). To determine if a function is a solution to a given differential equation, we do the same thing as before: evaluate the left-hand side of the d.e., evaluate the right-hand side of the d.e., and see if we get the same thing on each side. |