Find the exact solutions to the following mechanical vibration differential equa

Question

Find the exact solutions to the following mechanical vibration differential equations with initial conditions x(0) = 1 and x'(0) = 3; also classify as either overjumped, underdamped or critically damped and graph. 4x'+12x'+9x = 0

Explanation / Answer

Write the characteristic equation for this diff eq first 4r^2+12r+9=0 ==> (2r+3)^2=0 ==> r=-3/2 is a double root of the equation, corresponding to the eneral solution: x(t)=c1e^(-3/2t)+c2te^(-3/2t) To solve for the particular solution given the initial values, first plug in 0 for t to get: c1=1 Then take the derivative to get x'(t)=-3/2e^(-3/2t)+c2e^(-3/2t)-3/2c2te^(-3/2t) Plug in the given point to get 3=1+c2 c2=2 Therefore, the exact solution is: x(t)=e^(-3/2t)+2te^(-3/2t) To figure out the oclscillation status, consider the original equation. Divide it by four to get x"+3x'+9/4x=0 (w0)^2=9/4 (the coefficient in front of x) w0=3/2 w0*Y=3 (Y is the coefficient of damping, 3 is the coefficient in front of x') Y=3/2 Because Y>1, the system is overdamped. This can easily be graphed on any graphing utility. The key thing to note is the system does not oscillate and asymptotically approaches zero.

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