Question

In Exercises 3-5, consider the system \frac{dx}{dt} = -x + 3y \frac{dy}{dt} = -3x - y. 3. Verify that $Y_1(t) = (e^{-t}\sin(3t), e^{-t}\cos(3t))$ is a solution of this system. 4. Verify that $Y_2(t) = (e^{-(t-1)}\sin(3(t-1)), e^{-(t-1)}\cos(3(t-1)))$ is a solution. 5. Using HPGSystemSolver, sketch the solution curves for $Y_1(t)$ and $Y_2(t)$ in the xy-phase plane. Why don't $Y_1(t)$ and $Y_2(t)$ contradict the Uniqueness Theorem?

          In Exercises 3-5, consider the system
\frac{dx}{dt} = -x + 3y
\frac{dy}{dt} = -3x - y.
3. Verify that $Y_1(t) = (e^{-t}\sin(3t), e^{-t}\cos(3t))$ is a solution of this system.
4. Verify that $Y_2(t) = (e^{-(t-1)}\sin(3(t-1)), e^{-(t-1)}\cos(3(t-1)))$ is a solution.
5. Using HPGSystemSolver, sketch the solution curves for $Y_1(t)$ and $Y_2(t)$ in the
xy-phase plane. Why don't $Y_1(t)$ and $Y_2(t)$ contradict the Uniqueness Theorem?

In Exercises 3-5, consider the system
(dx)/(dt) = -x + 3y
(dy)/(dt) = -3x - y.
3. Verify that Y1(t) = (e^-tsin(3t), e^-tcos(3t)) is a solution of this system.
4. Verify that Y2(t) = (e^-(t-1)sin(3(t-1)), e^-(t-1)cos(3(t-1))) is a solution.
5. Using HPGSystemSolver, sketch the solution curves for Y1(t) and Y2(t) in the
xy-phase plane. Why don't Y1(t) and Y2(t) contradict the Uniqueness Theorem?

Added by Douglas P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

02/22/2023

Step 1

Let x = e^(-t)sin(3t) and y = e^(-t)cos(3t). Substitute x and y into the first equation: (dx)/(dt) = d(e^(-t)sin(3t))/(dt) = -e^(-t)sin(3t) + 3e^(-t)cos(3t) = -x + 3y Substitute x and y into the second equation: (dy)/(dt) = d(e^(-t)cos(3t))/(dt) = Show more…

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In Exercises 3-5, consider the system: (dx)/(dt) = -x + 3y (dy)/(dt) = -3x - y Verify that Y₁(t) = (e^(-t)sin(3t), e^(-t)cos(3t)) is a solution of this system. Verify that Y₂(t) = (e^(-(t-1))sin(3(t-1)), e^(-(t-1))cos(3(t-1))) is a solution. Using HPGSystemSolver, sketch the solution curves for Y₁(t) and Y₂(t) in the xy-phase plane. Why don't Y₁(t) and Y₂(t) contradict the Uniqueness Theorem? In Exercises 3-5, consider the system: (dx)/(dt) = -x + 3y (dy)/(dt) = -3x - y 3. Verify that Y₁(t) = e^(-sin3t), e^(-cos3t) is a solution of this system. 4. Verify that Y₂(t) = (e-t-1)sin3(t-1), e-t-1cos(3t-1) is a solution. 5. Using HPGSystemSolver, sketch the solution curves for Y₁(t) and Y₂(t) in the xy-phase plane. Why don't Y₁(t) and Y₂(t) contradict the Uniqueness Theorem?