Podcast
Questions and Answers
General solution for the homogeneous equation is given by xg (t) = ______.
General solution for the homogeneous equation is given by xg (t) = ______.
Ce^{-at}
The solution that satisfies the zero initial condition is x(t) = ______.
The solution that satisfies the zero initial condition is x(t) = ______.
0
The solution that satisfies the non-zero initial condition is x(t) = ______.
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) = ______.
The full solution to Equation (1) can finally be written as x(t) = ______.
By exponentiation on both sides, the above expression reads |x| = ______ for C2 = eC1 > 0.
By exponentiation on both sides, the above expression reads |x| = ______ 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 ______.
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 ______.
The full solution to Equation (1) can finally be written as x(t) = ______.
The full solution to Equation (1) can finally be written as x(t) = ______.
the transient solution involves $e^{-mt}$, in the limit $t\to\infty$ this term ______ out
the transient solution involves $e^{-mt}$, in the limit $t\to\infty$ this term ______ out
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 ______
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 ______
$mg\sin\theta \lim_{t\to\infty} v(t) =$ ______
$mg\sin\theta \lim_{t\to\infty} v(t) =$ ______
Using the Laplace transform, the free response solution for the driven harmonic oscillation is x(t) = _______ cos(ωt) + _______ sin(ωt).
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).
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) = _______.
The terminal velocity solution for the equation mv̇ + cv = mg sin θ is v(t) = _______.
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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).
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