CloudKel said:Homework Statement
a) ẍ + 5ẋ + 4x = 0
x(0) = 0, ẋ(0)=1
What type of damping?
b) ẍ + x = cos(t)
x(0) = 0, ẋ(0)= 1
What type of motion?
The Attempt at a Solution
a) Let x = R Eᴿᵀ
R = -4
R = -1
x(t) = CE**-4T + DE**-t
C + D = 0
-4C - D = 1
C = 1/5
D = 1/5
And it is over damping
I think this is the right answer for a) but i have no idea how to do b)
A second order linear differential equation is a mathematical expression that involves a second derivative of an unknown function and can be written in the form y'' + p(x)y' + q(x)y = g(x), where p(x) and q(x) are functions of x and g(x) is a known function.
The general solution to a second order linear differential equation is a family of functions that satisfies the equation. It contains two arbitrary constants, as a second order equation requires two initial conditions to be fully determined.
To solve a second order linear differential equation, you can use various methods such as the method of undetermined coefficients, the method of variation of parameters, or the Laplace transform method. You can also use software or calculators to find the solution.
Second order linear differential equations are used in various fields of science and engineering, such as physics, chemistry, biology, economics, and engineering. They are particularly useful in modeling systems that involve acceleration, oscillations, or growth and decay phenomena.
Both first and second order linear differential equations involve derivatives of an unknown function, but a second order equation has a second derivative while a first order equation has a first derivative. Additionally, a second order equation requires two initial conditions to be solved, while a first order equation only requires one. The methods used to solve them are also different.