Solve the differential equation: #(d^2y)/ (dx^2) −8 (dy)/(dx) =−16y#? Discuss what kind of differential equation is this, and when it may arise?
best written as
which shows that this is linear second order homogeneous differential equation
it has characteristic equation
which can be solve as follows
this is a repeated root so the general solution is in form
this is non-oscillating and models some kind of exponential behaviour that really depends on the value of A and B. One might guess it could be an attempt to model population or predator/prey interaction but i can't really say anything very specific.
it shows instability and that's about all i could really say about it
The differential equation
is a linear homogeneous constant coefficient equation.
For those equations the general solution has the structure
Substituting we have
Solving we obtain
When the roots repeat,
So, to maintain the number of initial conditions, we include them as independent solutions.
In this case we have
which results in
Those equations appear when modelling linear lumped parameter systems like those found in linear circuit theory or linear mechanics. Those equations are normally handled using operational algebraic methods like Laplace Transform methods