Solving the Non-linear Differential Equation

We need to solve the first-order non-linear ordinary differential equation (ODE):

with the initial condition $y(0) = 0.5$. We aim to find the value of $y(3)$.

Separating Variables

This is a separable ODE. Rearranging the terms gives:

Factor the denominator:

Integration using Partial Fractions

Apply partial fraction decomposition to the left side:

Solving for A and B yields $A = -1$ and $B = 1$. Thus:

Integrate both sides:

This results in:

Combining the logarithms:

Applying the Initial Condition

Use $y(0) = 0.5$ to find the constant $C$:

The equation becomes:

Solving for y(x)

Exponentiating both sides:

Since $y(0) = 0.5$ is between 0 and 1, and the ODE's right side $y^2-y$ is negative for $0 < y < 1$, the solution $y(x)$ remains between 0 and 1. Therefore, $y-1$ is negative, and $\frac{y-1}{y}$ is negative. So, $|\frac{y-1}{y}| = -(\frac{y-1}{y}) = \frac{1-y}{y}$.

Rearrange to solve for $y$:

Calculating y(3)

Substitute $x = 3$ into the solution:

Using the approximate value $e^3 \approx 20.0855$:

Calculate the final value:

The value $y(3) \approx 0.04741$ is nearest to 0.047.