To find the solution curve to the differential equation dy/dx = -1.233(y - 80) with the initial condition y(0) = -40, we can follow these steps:
Step 1: Rewrite the differential equation in the form dy/dx = f(x, y).
dy/dx = -1.233(y - 80)
Step 2: Determine if the equation is separable or not.
The given equation is not separable.
Step 3: Solve the equation by using an integrating factor.
In this case, we can use the integrating factor method to solve the equation.
Multiply both sides of the equation by the integrating factor, which is e^(1.233x):
e^(1.233x) * dy/dx = -1.233(y - 80) * e^(1.233x)
This simplifies to:
(e^(1.233x) * dy) / (y - 80) = -1.233 * e^(1.233x) * dx
Integrate both sides:
∫ (e^(1.233x) * dy) / (y - 80) = ∫ -1.233 * e^(1.233x) * dx
This gives:
ln|y - 80| = -1 * e^(1.233x) + C
Where C is the constant of integration.
Step 4: Solve for y.
To solve for y, we can exponentiate both sides of the equation by raising e to the power of both sides:
|y - 80| = e^(-e^(1.233x) + C)
Taking the positive and negative cases, we have:
y - 80 = e^(-e^(1.233x) + C) or -(y - 80) = e^(-e^(1.233x) + C)
Simplify each case:
y = 80 + e^(-e^(1.233x) + C) or y = 80 - e^(-e^(1.233x) + C)
Step 5: Apply the initial condition to find the specific solution.
Given the initial condition y(0) = -40, we can substitute x = 0 and y = -40 into the general solution to find the specific solution.
In this case, let's consider the positive case:
y = 80 + e^(-e^(1.233x) + C)
When x = 0 and y = -40:
-40 = 80 + e^(-e^(0) + C) = 80 + e^C
Solving for C:
e^C = -120
Since e^C cannot be negative, there is no specific solution that satisfies the given initial condition.
Therefore, the statement "There is no solution curve to the differential equation with initial condition y(0) = -40" best describes the solution curve to the given differential equation with the given initial condition.