INSTRUCTOR: Find the general solution to the second derivative of y minus 4 times the first derivative of y plus 4 times y equals 7 sine t minus cosine t. First, we want to consider the complementary equation, which would be the second derivative of y minus 4 times the first derivative of y plus 4y equals 0.

The characteristic equation of this is lambda squared minus 4 lambda plus 4 equals 0. And this factors as parentheses, lambda minus 2, close parentheses, squared equals 0. So the lambda equals 2.

And this is a repeated solution, which means, based on what we know, the different forms for different types of solutions, that the general solution is-- general solution is c1 times e to the 2t exponent, end exponent, plus c sub 2 times t times e to the exponent of 2t, end exponent.

We'll want to use this as part of our final answer. Next, we look at the other side of the equation, the right-hand side. 7 sine t minus cosine t. Based on this, we want to guess that y sub p of t is equal to A cosine t plus B sine t.

Now, we're going to take this guess of a solution and evaluate it in it for y, into the equation we have. And ultimately, we'll be able to determine what the specific A and B values are, those parameters. All right, so the second derivative of this function minus 4 times the first derivative of that function plus 4 times that function equals 7 sine t minus cosine t.

So this is going to be parentheses, negative A cosine t minus B sine t, close parentheses, minus 4, open parentheses, negative A sine t plus B cosine t, close parentheses, plus 4, open parentheses, A cosine t plus B sine t, close parentheses, equals 7 sine t minus cosine t.

Case you're wondering where those values came from, we take the second derivative of y sub p to get the first set of parentheses, the first derivative of y sub p to get the second set of parentheses, and y sub p itself for the third. Now, we want to group these in sine terms and cosine terms so we can equate coefficients.

So if we combine these together, it's going to be equal to parentheses, negative B plus 4A plus 4B, close parentheses, sine t plus, open parentheses, negative A minus 4B plus 4A, close parentheses, cosine t. And this is equal to 7 sine t minus cosine t.

Here's where equating coefficients comes in. Equating coefficients for sine, we see that 4A plus 3B is equal to 7 and also that 3A minus 4B is equal to negative 1, equating coefficients for the cosine terms. Now, using elimination or some other method of solving this, we can see that A equals 1 and B equals 1.

So that means that our specific yet general solution for this is going to be y of t equals c sub 1 times e to the 2t exponent, end exponent, plus c sub 2 times t times e to the 2t, end exponent, plus cosine t plus sine t.