I get how a system of first of first order differential equations would work for the system of two tanks, but I'm having a hard time visualizng/conceptualizing how a second order system would work, any help on that part of the problem would be greatly appreciated! **Second-Order Homogeneous Equation Example****Problem Statement:**1. **Equation Example:** - Provide an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation.2. **Corresponding Linear System:** - Write down the linear system corresponding to this differential equation.3. **Real-World Application:** - Find an example of an interconnected system of two tanks that can be described by this system. Explain the relationship.4. **Solution:** - Solve the linear system described.**Details and Explanation:**- **Equation Example:** An example of such an equation is: \[ ay'' + by' + cy = 0 \] Example with non-real roots: \[ y'' + 2y' + 5y = 0 \]- **Characteristic Equation:** The characteristic equation is: \[ r^2 + 2r + 5 = 0 \] Solving gives non-real roots \( r = -1 \pm 2i \).- **Corresponding Linear System:** The equivalent system can be written as: \[ \begin{align*} x_1' &= x_2 \\ x_2' &= -5x_1 - 2x_2 \end{align*} \] This is derived by setting \( x_1 = y \) and \( x_2 = y' \).- **Real-World Application:** Consider two interconnected tanks where the rate of change of the liquid level in each tank can be modeled by the given system of equations. The liquid exchange between the tanks can mimic the behavior of the system described by the differential equation.- **Solution of the System:** To solve, use: \[ x_1(t) = e^{-t} (C_1 \cos 2t + C_2 \sin 2t) \] \[ x_2(t) = e^{-t} ((-2C_1 + C_2)\cos 2t + (-C_1 - 2C_2)\sin 2t) \] where \( C_1 \) and \( C_2 \) are constants determined by initial

a second order, constant coefficient, homogenous equation will look like ay''+by'+cy=0 consider an…

Give an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation. Write down the linear system that corresponds to it. Can you find an example of an interconnected system of two tanks that is described by this system? Explain. Then solve the system.

Give an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation. Write down the linear system that corresponds to it. Can you find an example of an interconnected system of two tanks that is described by this system? Explain. Then solve the system.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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