Suppose that y = f (t) satisfies the differential equation dy/dt=y(2−y) and initial condition f (0) = 1. Find an explicit expression for f (t).
To find an explicit expression for f(t) given the differential equation dy/dt = y(2 - y) and the initial condition f(0) = 1, we can solve the differential equation using separation of variables.
Step 1: Rewrite the differential equation
dy/dt = y(2 - y)
Step 2: Separate the variables
dy / (y(2 - y)) = dt
Step 3: Integrate both sides
∫(1 / (y(2 - y))) dy = ∫dt
To integrate the left-hand side, we can use partial fraction decomposition. Let's find the partial fraction decomposition of 1 / (y(2 - y)):
1 / (y(2 - y)) = A / y + B / (2 - y)
Multiplying both sides by y(2 - y), we get:
1 = A(2 - y) + By
Expanding and rearranging, we have:
1 = (2A - A)y + (B - Ay)
Equating coefficients, we get:
2A - A = 0 => A = 1
B - A = 0 => B = 1
So, the partial fraction decomposition becomes:
1 / (y(2 - y)) = 1 / y + 1 / (2 - y)
Now, let's integrate both sides:
∫(1 / y + 1 / (2 - y)) dy = ∫dt
Step 4: Evaluate the integrals
∫(1 / y) dy = ln|y| + C1
∫(1 / (2 - y)) dy = -ln|2 - y| + C2
where C1 and C2 are constants of integration.
So, we have:
ln|y| - ln|2 - y| = t + C
Step 5: Simplify the equation
Using the logarithmic property ln(a) - ln(b) = ln(a / b), we can rewrite the equation as:
ln|y / (2 - y)| = t + C
Step 6: Apply the initial condition
Given that f(0) = 1, we substitute t = 0 and y = 1 into the equation:
ln|1 / (2 - 1)| = 0 + C
ln(1) = C
C = 0
So, the equation becomes:
ln|y / (2 - y)| = t
Step 7: Solve for f(t)
Taking the exponential of both sides, we get:
|y / (2 - y)| = e^t
Since the absolute value |a/b| = |a| / |b|, we have:
y / (2 - y) = ±e^t
Solving for y, we have two cases:
Case 1: y / (2 - y) = e^t
y = (2 - y)e^t
y = 2e^t - ye^t
y + ye^t = 2e^t
y(1 + e^t) = 2e^t
y = (2e^t) / (1 + e^t)
Case 2: y / (2 - y) = -e^t
y = -(2 - y)e^t
y = -2e^t + ye^t
2e^t + ye^t = y
2e^t = y(1 - e^t)
y = (2e^t) / (1 - e^t)
Therefore, the explicit expression for f(t) is:
f(t) = (2e^t) / (1 + e^t) or f(t) = (2e^t) / (1 - e^t)