Solved Exercises

Let's solve our first example. We have the equation d²y/dx² + 7 dy/dx + 12y = 0. To start, we need to find the auxiliary equation.

How do we derive the auxiliary equation from this differential equation?

Good question! The auxiliary equation is derived by replacing d²y/dx² with m², dy/dx with m, and y with 1. So it becomes m² + 7m + 12 = 0. Now we can use the quadratic formula.

I see, so we get m = -3 and m = -4, correct?

Exactly! This gives us distinct real roots. Thus, the general solution will be y(x) = C₁e^{-3x} + C₂e^{-4x}.

What does this solution represent in a physical context?

This solution is indicative of exponential decay, commonly found in processes like damped vibrations. Remember, distinct roots indicate a unique behavior in our solutions.

So the key takeaway here is how to identify and interpret the roots of an auxiliary equation for a second-order differential equation. Let's summarize: the auxiliary equation helps derive the general solution, and the nature of roots tells us about the system's behavior.

Now, let's move on to our second exercise. Our equation here is d²y/dx² + 6dy/dx + 9y = 0. Can anyone tell me the form of the auxiliary equation?

It would be m² + 6m + 9 = 0, right?

Yes! Now let's solve it. What do we get?

I think we get a repeated root at m = -3.

Exactly! In this case, since we have a repeated root, how would the general solution look?

It should be y(x) = (C₁ + C₂x)e^{-3x}, since we have to account for the multiplicity.

Correct! This solution implies that the system's response will decay exponentially, but with a linear modification due to repeated roots.

To recap, when we encounter repeated roots, we add an x term to our constants in the general solution. It signifies a different interaction compared to distinct roots.

Let's finish with the last exercise involving complex roots. Our equation is d²y/dx² + 16y = 0. What can we do when we see this form?

We can rewrite it as m² + 16 = 0 to find the roots.

Exactly! Can someone tell me what the roots look like?

I think it gives us m = ±4i, which are complex roots.

Correct! With complex roots, how do we form our general solution?

It would be in the form y(x) = e^{0x}(C cos(4x) + C sin(4x)) because alpha is zero in this case.

Right! This form shows oscillatory behavior typical in mechanical vibrations. The solution represents sinusoidal motion, which is crucial in engineering applications.

The take-home message is that complex roots yield oscillatory solutions, highly relevant for studying systems like oscillators or waves.