Consider the following initial value problem.

dy
dx
   =  (y2 − 5y + 4) sin2( 
2πy
7
 ) ,    y(0) = a.

Give a possible value of the real number a for which the solution to the corresponding initial value problem is a non-constant function that satisfies

lim
x→∞
 y(x)  =   
7
2
 

We are given the differential equation

\[
\frac{dy}{dx} = (y^2 - 5y + 4) \sin^2 \left( \frac{2\pi y}{7} \right)
\]

with the initial condition \( y(0) = a \), and we are asked to find a possible value of \( a \) such that the solution to this problem is a non-constant function that satisfies

\[
\lim_{x \to \infty} y(x) = \frac{7}{2}.
\]

### Step 1: Identify Equilibrium Points

The right-hand side of the differential equation is given by

\[
f(y) = (y^2 - 5y + 4) \sin^2 \left( \frac{2\pi y}{7} \right).
\]

For equilibrium points, we need to solve for values of \( y \) such that \( f(y) = 0 \). This can happen if either:

1. \( y^2 - 5y + 4 = 0 \), or
2. \( \sin^2 \left( \frac{2\pi y}{7} \right) = 0 \).

#### Case 1: \( y^2 - 5y + 4 = 0 \)

Solve the quadratic equation:

\[
y^2 - 5y + 4 = (y - 1)(y - 4) = 0.
\]

Thus, \( y = 1 \) or \( y = 4 \) are equilibrium points.

#### Case 2: \( \sin^2 \left( \frac{2\pi y}{7} \right) = 0 \)

This implies \( \sin \left( \frac{2\pi y}{7} \right) = 0 \), which occurs when

\[
\frac{2\pi y}{7} = n\pi, \quad n \in \mathbb{Z}.
\]

Thus, \( y = \frac{7n}{2} \). The values of \( y \) that satisfy this equation include \( y = 0, \frac{7}{2}, 7, \dots \).

### Step 2: Behavior of the Solution

We know that the solution satisfies \( \lim_{x \to \infty} y(x) = \frac{7}{2} \), which suggests that \( y = \frac{7}{2} \) is an attractive equilibrium point for the solution.

### Step 3: Analyze the Initial Condition

To ensure that the solution tends toward \( y = \frac{7}{2} \) as \( x \to \infty \), we need to choose the initial value \( a \) such that the solution starts near \( y = \frac{7}{2} \) but does not settle at any other equilibrium point (such as \( y = 1 \) or \( y = 4 \)).

Since \( y = 1 \) and \( y = 4 \) are also equilibrium points, and we want the solution to approach \( \frac{7}{2} \) as \( x \to \infty \), a reasonable choice for \( a \) is a value close to, but not equal to, \( \frac{7}{2} \).

### Step 4: Conclusion

A possible value of \( a \) that would lead to a non-constant solution tending to \( y = \frac{7}{2} \) is \( a = 4 \). This is because starting near \( y = 4 \) allows the solution to move towards the attractive equilibrium point \( \frac{7}{2} \) without remaining at \( y = 4 \) (as \( y = 4 \) is an unstable equilibrium for this problem).

### Final Answer:

A possible value of \( a \) is \( a = 4 \).

---

Would you like more details or have any further questions?

### Related Questions:

1. How do we analyze the stability of equilibrium points in a differential equation?
2. What is the significance of the function \( \sin^2 \left( \frac{2\pi y}{7} \right) \) in this context?
3. Could \( a = 1 \) also lead to a solution that tends towards \( y = \frac{7}{2} \)?
4. How would the solution behave if \( a = 0 \)?
5. Can we have a solution where \( y(x) \) oscillates between equilibrium points?

### Tip:

In problems involving equilibrium points, analyzing their stability is key to understanding the long-term behavior of the solutions.

Consider the following initial value problem. dy/dx = (y^2 − 5y + 4) sin^2(2πy/7), y(0) = a. Give a possible value of the real number a for which the solution to the corresponding initial value problem is a non-constant function that satisfies lim (x→∞) y(x) = 7/2.

Explore how to solve an initial value problem involving the differential equation dy/dx = (y^2 - 5y + 4) sin^2(2πy/7). Learn to find the value of a such that the solution satisfies lim (x→∞) y(x) = 7/2, using concepts from differential equations, equilibrium analysis, and sinusoidal functions. Suitable for undergraduate calculus students.

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