The differential equation $$y''+3y'-28y=0$$ has a characteristic equation with two distinct real solutions. What is the larger of these two solutions?

Question

BlackTom AI · Accepted Answer

We begin by converting the differential equation into its characteristic equation. For a linear constant-coefficient ODE of the form y'' + ay' + by = 0, the characteristic equation is r^2 + ar + b = 0. Here, a = 3 and b = -28, so the characteristic equation is r^2 + 3r - 28 = 0.

Next, factor the quadratic if possible. We look for two numbers that multiply to -28 and sum to 3. The numbers 7 and -4 fit because 7 * (-4) = -28 and 7 + (-4) = 3.

Using these factors, we factor the quadratic as (r + 7)(r - 4) = 0.

Setting each factor equal to zero yields the two roots: r = -7 and r = 4. These are real and distinct, as the problem states. Among these two roots, the larger one is the one with the greater numerical value, which is 4.

To recap the process: write the characteristic equation, factor it to find the roots, and compare the roots to identify the larger among them.

The differential equation \[y''+3y'-28y=0\] has a characteristic equation with two distinct real solutions. What is the larger of these two solutions?Short answer

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