2.9. Exercises

1. Solve

d²y/dx² + 7 dy/dx + 12y = 0

This exercise requires us to solve a second-order homogeneous linear differential equation. The equation can be identified by its structure, as it involves second derivatives of y with respect to x. To solve it, we first derive its auxiliary or characteristic equation by assuming a solution of the form y = e^(mx), where m is a constant. By substituting this into the differential equation, we get a polynomial in m, which we solve to find the roots. The nature of these roots will determine the form of the general solution.

Imagine a bridge swaying slightly in the wind. The way it moves can be predicted using similar differential equations, as the bridge acts like a system responding to external forces, and by solving these kinds of equations, engineers can understand how to design stronger structures.

2. Solve

d²y/dx² + 6 dy/dx + 9y = 0

In this problem, we again start with the second-order differential equation. By forming the auxiliary equation, we will identify the coefficients for m², m, and the constant term. Solving the auxiliary equation reveals whether the roots are real or repeated. If we find repeated roots, this helps us understand that the general solution will include terms of x in addition to the exponential part due to the multiplicity of the roots.

Consider a child on a swing; if two children sit on opposite sides, they can cause the swing to oscillate back and forth. The equations governing the swing’s motion are modeled by similar differential equations, helping us predict how long it will sway and how quickly it will come to rest.

3. Solve

d²y/dx² - 10y = 0

This exercise introduces a different structure of a second-order differential equation. By rearranging the terms, we again form the auxiliary equation and solve for m. This time, we expect real roots indicating exponential growth or decay in our solutions. Understanding the implications of the solution's nature helps in analysis and prediction of system behavior.

Think of an alarm system that goes off based on certain thresholds. If we can model how quickly it activates or deactivates under different conditions using equations like this, we can better engineer out systems to prevent false alarms or ensure effectiveness.

4. A structure’s vibration is modeled as

d²y/dx² + 16y = 0. Find its solution.

In this exercise, we need to solve for the vibrations modeled by a differential equation with a specific structure. The key process remains the same: derive the auxiliary equation and solve for roots which are complex in this case. Their interpretation leads us to solutions involving sine and cosine functions indicative of oscillatory motion.

Imagine a guitar string being plucked. The vibration patterns can be predicted using these types of formulas, allowing the instrument maker to craft strings that deliver the desired tones by understanding their vibration frequencies.

5. Prove that the general solution of a second-order linear homogeneous ODE always contains two arbitrary constants.

This exercise asks you to consider the properties of second-order linear homogeneous ordinary differential equations (ODEs). Their solutions can be expressed generally with two arbitrary constants because these equations typically result in a second-degree polynomial in their auxiliary equation. This leads to two roots, each representing a different exponential function in the solution that can be scaled (shifted) by these constants.

Think about a pair of custom-fit shoes; just as they can be adjusted to fit any foot size (the constants), the general solutions in these equations can be shaped to match initial conditions in physical systems, giving engineers the flexibility to tailor solutions to specific problems.

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