A dynamic system is represented by a second order linear differential equation. 2frac{d^2x}{dt^2}+5frac{dx}{dt}-3x=0 The initial conditions are given as: when t=0, x=4 and frac{dx}{dt}=9 Solve the differential equation and obtain the output of the system x(t) as afunction of t.

Khadija Wells 2021-02-16 Answered
A dynamic system is represented by a second order linear differential equation.
2d2xdt2+5dxdt3x=0
The initial conditions are given as:
when t=0, x=4 and dxdt=9
Solve the differential equation and obtain the output of the system x(t) as afunction of t.
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faldduE
Answered 2021-02-17 Author has 109 answers
Solution. Given 2d2xdt2+5dxdt3x=0 (1)
when t=0 then x=4 and dxdt=9
Auxiliary equation
2m2+5m3=0
2m2+6mm3=0
(2m1)(m+3)=0
m1=3 and m2=12
Auxiliary equation
Then solution x(t)=c1e3t+c2e12t (2)
dxdt=3c1e3t+12c2et2
x(0)=4
c1+c2=4 (3)
Also dxdt=9 when t=0
3c1+c22=9
6c1+c2=18 (4)
Solving (3) and (4) we get
5c1=14c1=145
And c2=65
Then solution (2) will becomes.
x(t)=145e3t+65et2
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