Second-order equations, Constant coefficient, Homogeneous equation, Characteristic equation, Solution of characteristic equation, Complex solution, Homogeneous solution.
Second-order homogeneous linear differential equations with constant coefficients
Here we have this equation.
Using the previous methods we can’t expect much success in solving this equation, since it is of second order.
Well, this is not a very encouraging outlook on the solution.
Equations like this are usually darn hard to solve.
But luckily, this type is an exception.
Let’s see what we should do with it.
This is the general form of the equation. The solution of such equations is always some
Let’s substitute this into the equation, and see what happens.
This equation is called the characteristic equation.
In order to solve the differential equation, we have to solve this quadratic equation.
The solution of the differential equation is:
If the characteristic equation has two different real solutions: and , then
If the characteristic equation has one real solution, then
If the characteristic equation has two different complex solutions:
And now, let's see the solution.
The characteristic equation is:
It seems we solved this one. Let's see another one.
Here comes the characteristic equation:
This wasn’t too hard either.
Finally let’ see the third type.
Well, there is a little trouble here.
There is a negative number under the root symbol, which means there is no real solution for the characteristic equation.
However, there is a complex solution. All we need to know is this:
And now let's see the solution.
It gets really exciting if the equation is nonhomogeneous.
Let’ see what happens in that case.