4.1.2

Speaker: We need to verify that the function y equals e to the negative 3x plus 2x plus 3 is a solution to the differential equation given y prime plus 3y equals 6x plus 11. To do this, first we need to determine the derivative of this function so that we can substitute all of those values into our differential equation. The derivative of e to the negative 3x would be negative 3e to the negative 3x plus 2. 

Now just plugging those in or substituting those, we would have y prime negative 3e to the negative 3x plus 2 plus 3 times y. So that is the-- let me put this term here, e to the negative 3x plus 2x plus 3 equals 6x plus 11. 

Distributing, we see negative 3e to the negative 3x plus 2 plus 3e to the negative 3x plus 6x plus 9 equals 6x plus 11. We see a few terms that we can eliminate. These two are a zero pair. So we are left with 2 plus 9-- so that's 11-- and a 6x term, which is equal to 6x plus 11. Because the value of y that's given, that function y equals, because that causes our differential equation to be true, it is indeed a solution to that differential equation.