13 Questions
General solution for the homogeneous equation is given by xg (t) = ______.
Ce^{-at}
The solution that satisfies the zero initial condition is x(t) = ______.
0
The solution that satisfies the non-zero initial condition is x(t) = ______.
Ce^{-at}
The full solution to Equation (1) can finally be written as x(t) = ______.
x(0)e^{-at} + \dfrac{b},{a}(1 - e^{-at})
By exponentiation on both sides, the above expression reads |x| = ______ for C2 = eC1 > 0.
$|x| = C2 e^{-at}$
For every prescribed initial condition x(0), the solution of Equation (1) may be written as a linear superposition of its general solution xg and particular solution ______.
xp
The full solution to Equation (1) can finally be written as x(t) = ______.
x(t) = xg(t) + xp(t)
the transient solution involves $e^{-mt}$, in the limit $t\to\infty$ this term ______ out
drops
the second term is interpreted as the steady state solution because as the transient term dies off, the velocity $v(t)$ approaches the terminal velocity $mg\cos\theta$, which is a ______
constant
$mg\sin\theta \lim_{t\to\infty} v(t) =$ ______
$c$
Using the Laplace transform, the free response solution for the driven harmonic oscillation is x(t) = _______ cos(ωt) + _______ sin(ωt).
x(t) = _______ is the forced response solution for the driven harmonic oscillation with zero initial conditions and f(t) = us(t).
The terminal velocity solution for the equation mv̇ + cv = mg sin θ is v(t) = _______.
Study Notes
Homogeneous Equation Solution
- The general solution to the homogeneous equation is xg(t) = Ce^(mt) + De^(-mt).
- The solution that satisfies the zero initial condition is x(t) = Ce^(mt).
Non-Zero Initial Condition Solution
- The solution that satisfies the non-zero initial condition is x(t) = Ce^(mt) + De^(-mt) + x_p(t).
Full Solution to Equation (1)
- The full solution to Equation (1) can be written as x(t) = Ce^(mt) + De^(-mt) + x_p(t).
Exponentiation and Initial Conditions
- By exponentiation on both sides, |x| = Ce^(mt) for C² = eC¹ > 0.
- For every prescribed initial condition x(0), the solution of Equation (1) may be written as a linear superposition of its general solution xg and particular solution xp.
Transient and Steady State Solutions
- The transient solution involves e^(-mt), which dies out in the limit t → ∞.
- The second term is interpreted as the steady-state solution because as the transient term dies off, the velocity v(t) approaches the terminal velocity mgcosθ.
Velocity and Terminal Velocity
- mg sinθ lim_(t→∞) v(t) = mg sinθ.
- The terminal velocity solution for the equation mv̇ + cv = mg sin θ is v(t) = (mg sinθ)/c.
Driven Harmonic Oscillation
- Using the Laplace transform, the free response solution for the driven harmonic oscillation is x(t) = A cos(ωt) + B sin(ωt).
- The forced response solution for the driven harmonic oscillation with zero initial conditions and f(t) = uₛ(t) is x(t) = (F₀/ω) sin(ωt).
Test your knowledge on time-domain methods for first order systems with this quiz. Covering topics such as inhomogeneous linear ordinary differential equations and the canonical form, this quiz will help you strengthen your understanding of time-domain treatment.
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