If y=e5t is a solution to the differential equation

d2ydt2−11dydt+ky=0,

find the value of the constant k and the general solution to this equation.

k=
y=

k = 30,

Step-by-step explanation:

Since is a solution, then it must satisfy the differential equation. So, we calculate the derivatives and replace the value in the equation. We have that

Then, replacing the derivatives in the equation we have:

Since is a positive function, we have that

.

Now, consider a general solution , then, by calculating the derivatives and replacing them in the equation, we get

We already know that r=5 is a solution of the equation, then we can divide the polynomial by the factor (r-5) to the get the other solution. If we do so, we get that (r-6)=0. So the other solution is r=6.

Therefore, the general solution is