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Consider the differential equation d ^ 2 * x / d * t ^ 2 + 2 * d/dt (x) + x = 0 At time t = 0 is given that x = 1 and d/dt (x) = 0 Att t = 1 , the value of x is given by
[GATE 2003]
(a)1/e
(b) 2/e
(c)1
(d)3/e?
[GATE 2003]
(a)1/e
(b) 2/e
(c)1
(d)3/e?
Most Upvoted Answer
Consider the differential equation d ^ 2 * x / d * t ^ 2 + 2 * d/dt (x...
Understanding the Differential Equation
The given differential equation is:
- d^2x/dt^2 + 2(dx/dt) + x = 0
This is a second-order linear homogeneous differential equation with constant coefficients.
Characteristic Equation
To solve this, we first form the characteristic equation:
- m^2 + 2m + 1 = 0
This factors to:
- (m + 1)^2 = 0
Thus, we have a double root:
- m = -1
General Solution
The general solution for the equation with a double root is given by:
- x(t) = (C1 + C2 * t) * e^(-t)
Where C1 and C2 are constants determined by initial conditions.
Applying Initial Conditions
Given initial conditions are:
- x(0) = 1
- dx/dt(0) = 0
1. For x(0) = 1:
- x(0) = (C1 + C2 * 0) * e^0 = C1 = 1
2. For dx/dt(0) = 0:
- dx/dt = (C2 - (C1 + C2 * t)) * e^(-t)
- At t = 0: 0 = (C2 - C1) * e^0 => C2 - 1 = 0 => C2 = 1
Thus, the solution becomes:
- x(t) = (1 + t) * e^(-t)
Finding x(1)
Now, we calculate x(1):
- x(1) = (1 + 1) * e^(-1) = 2/e
Final Answer
Therefore, the value of x at t = 1 is:
- (b) 2/e
The given differential equation is:
- d^2x/dt^2 + 2(dx/dt) + x = 0
This is a second-order linear homogeneous differential equation with constant coefficients.
Characteristic Equation
To solve this, we first form the characteristic equation:
- m^2 + 2m + 1 = 0
This factors to:
- (m + 1)^2 = 0
Thus, we have a double root:
- m = -1
General Solution
The general solution for the equation with a double root is given by:
- x(t) = (C1 + C2 * t) * e^(-t)
Where C1 and C2 are constants determined by initial conditions.
Applying Initial Conditions
Given initial conditions are:
- x(0) = 1
- dx/dt(0) = 0
1. For x(0) = 1:
- x(0) = (C1 + C2 * 0) * e^0 = C1 = 1
2. For dx/dt(0) = 0:
- dx/dt = (C2 - (C1 + C2 * t)) * e^(-t)
- At t = 0: 0 = (C2 - C1) * e^0 => C2 - 1 = 0 => C2 = 1
Thus, the solution becomes:
- x(t) = (1 + t) * e^(-t)
Finding x(1)
Now, we calculate x(1):
- x(1) = (1 + 1) * e^(-1) = 2/e
Final Answer
Therefore, the value of x at t = 1 is:
- (b) 2/e
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Consider the differential equation d ^ 2 * x / d * t ^ 2 + 2 * d/dt (x) + x = 0 At time t = 0 is given that x = 1 and d/dt (x) = 0 Att t = 1 , the value of x is given by[GATE 2003](a)1/e(b) 2/e(c)1(d)3/e?
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