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Q4. Let k≠0k \neq 0, and let q(t)q(t) be continuous and such that \[ \lim _{t \ri...
Feb 1, 2024
Q4. Let k≠0k \neq 0, and let q(t)q(t) be continuous and such that lim⁡t→∞q(t)=0,∫0∞e−k2sq(s)ds=0. \lim _{t \rightarrow \infty} q(t)=0, \quad \int_{0}^{\infty} e^{-k^{2} s} q(s) d s=0 . Show that the solution y(t)y(t) of the IVP y′(t)−k2y(t)=q(t),y(0)=y0 y^{\prime}(t)-k^{2} y(t)=q(t), y(0)=y_{0} tend to zero as t→∞t \rightarrow \infty if and only if y0=0y_{0}=0.
Solution by Steps
step 1
Consider the given differential equation and initial condition: y′(t)−k2y(t)=q(t),y(0)=y0 y'(t) - k^2 y(t) = q(t), y(0) = y_0
step 2
Take the Laplace transform of both sides of the differential equation: L{y′(t)}−k2L{y(t)}=L{q(t)} \mathcal{L}\{y'(t)\} - k^2 \mathcal{L}\{y(t)\} = \mathcal{L}\{q(t)\}
step 3
Use the properties of the Laplace transform to simplify: sL{y(t)}−y0−k2L{y(t)}=L{q(t)} s\mathcal{L}\{y(t)\} - y_0 - k^2 \mathcal{L}\{y(t)\} = \mathcal{L}\{q(t)\}
step 4
Solve for L{y(t)} \mathcal{L}\{y(t)\} : L{y(t)}=L{q(t)}+y0s+k2 \mathcal{L}\{y(t)\} = \frac{\mathcal{L}\{q(t)\} + y_0}{s + k^2}
step 5
Take the inverse Laplace transform to find y(t) y(t) : y(t)=L−1{L{q(t)}+y0s+k2} y(t) = \mathcal{L}^{-1}\left\{\frac{\mathcal{L}\{q(t)\} + y_0}{s + k^2}\right\}
step 6
Apply the final value theorem to determine the limit as t→∞ t \rightarrow \infty : lim⁡t→∞y(t)=lim⁡s→0sL{y(t)} \lim_{t \rightarrow \infty} y(t) = \lim_{s \rightarrow 0} s \mathcal{L}\{y(t)\}
step 7
Substitute the expression for L{y(t)} \mathcal{L}\{y(t)\} into the limit: lim⁡s→0s(L{q(t)}+y0s+k2) \lim_{s \rightarrow 0} s \left(\frac{\mathcal{L}\{q(t)\} + y_0}{s + k^2}\right)
step 8
Since lim⁡t→∞q(t)=0 \lim_{t \rightarrow \infty} q(t) = 0 , we have lim⁡s→0sL{q(t)}=0 \lim_{s \rightarrow 0} s \mathcal{L}\{q(t)\} = 0
step 9
Evaluate the limit: lim⁡s→0sy0s+k2=0⋅y00+k2=0 \lim_{s \rightarrow 0} \frac{s y_0}{s + k^2} = \frac{0 \cdot y_0}{0 + k^2} = 0 , provided that y0=0 y_0 = 0
step 10
Conclude that y(t) y(t) tends to zero as t→∞ t \rightarrow \infty if and only if y0=0 y_0 = 0
Answer
The solution y(t) y(t) of the IVP tends to zero as t→∞ t \rightarrow \infty if and only if y0=0 y_0 = 0 .
Key Concept
Final Value Theorem and Laplace Transform
Explanation
The final value theorem, in conjunction with the Laplace transform, is used to determine the behavior of y(t) y(t) as t t approaches infinity. The given conditions on q(t) q(t) and the initial value y0 y_0 are crucial in determining the long-term behavior of the solution to the differential equation.
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