Show that y = 5x2 + 2 is a solution to the d.e.
We know that we should get the same thing when we evaluate the two sides of the differential equation. If y = 5x2 + 2 theny' = 10x and y" = 10. The left-hand side of the d.e. is
and the right-hand side of the d.e. is
2y = 2(5x2 + 2) = 10x2 + 4.
Since we got the same thing on both sides of the d.e., we have shown that y = 5x2 + 2 is a solution to this differential equation.