How would I solve the following equation d^2x/dt^2 + 5dx/dt + 6x = 0

Our given equation is d²x/dt² + 5dx/dt + 6x = 0, which we need to recognise as a second order differential equation. Therefore we need to begin by solving the auxilary funtion m²+5m +6= 0. ( Side note: Most of the mathematical equations we solve are expressed in x and y, but in this equation it's expressed in terms of x and t, where x is the dependent variable). Solving the auxiliary funtion gives us values of -3&-2 for m. Because these are real values that are not equal to each other we can use the complimentary funtion y= Ae^ct + Be^dt where y is the dependent variable, t is our independent variable and A&B are constants of intergration. If we plug in our values the auxiliary funtion becaomes x = Ae^-3t+Be^-2t. Which is our final answer.