Solve the initial initial value problem
d^2x/dt^2+2DX/dt-3x =0, x(2π) = 0, x(2π) = 13
To solve the initial value problem, we need to find the function x(t) that satisfies the given second-order linear differential equation and the initial conditions x(2π) = 0 and x'(2π) = 13.
Step 1: Finding the Characteristic Equation
The given differential equation is:
d^2x/dt^2 + 2D(dx/dt) - 3x = 0
To solve this equation, we assume a solution of the form x(t) = e^(rt), where r is a constant. Substituting this into the differential equation, we get:
r^2e^(rt) + 2Dr(e^(rt)) - 3e^(rt) = 0
We can now factor out e^(rt):
e^(rt)(r^2 + 2Dr - 3) = 0
For this equation to hold true for all values of t, we must have:
r^2 + 2Dr - 3 = 0
This is the characteristic equation of the differential equation.
Step 2: Solving the Characteristic Equation
To solve the characteristic equation, we can use the quadratic formula:
r = (-2D ± √(4D^2 + 12)) / 2
Simplifying this expression, we get:
r = -D ± √(D^2 + 3)
Therefore, the solutions for r are:
r1 = -D + √(D^2 + 3) and r2 = -D - √(D^2 + 3)
Step 3: Finding the General Solution
The general solution of the given differential equation is of the form:
x(t) = c1e^(r1t) + c2e^(r2t)
where c1 and c2 are arbitrary constants to be determined from the initial conditions.
Step 4: Applying the Initial Conditions
We are given the initial conditions x(2π) = 0 and x'(2π) = 13.
Using the first initial condition, we can substitute t = 2π into the general solution:
0 = c1e^(r1(2π)) + c2e^(r2(2π))
Using the second initial condition, we can differentiate the general solution with respect to t and substitute t = 2π:
13 = c1(r1e^(r1(2π)) + r2e^(r2(2π))) + c2(r1e^(r1(2π)) + r2e^(r2(2π)))
These two equations give a system of linear equations in c1 and c2. We can solve this system to find the values of c1 and c2.
Step 5: Solve the System of Equations
Solve the system of equations:
0 = c1e^(2πr1) + c2e^(2πr2)
13 = c1(r1e^(2πr1) + r2e^(2πr2)) + c2(r1e^(2πr1) + r2e^(2πr2))
where r1 = -D + √(D^2 + 3) and r2 = -D - √(D^2 + 3).
By solving this system of equations, you can find the values of c1 and c2.
Step 6: Write the Final Solution
Once you have determined the values of c1 and c2, substitute them back into the general solution to obtain the final solution.
x(t) = c1e^(r1t) + c2e^(r2t)
where c1 and c2 are the values you found in the previous step.