Which of the following statements best describes the solution curve to the differential equation dy/dx=-1.233(y-80), with initial condition y(0)=-40?
It's C:
The solution curve y=y(x) is modified exponential decay, and y -> 80 as x -> ∞
A. The solution curve y=y(x) is modified exponential growth, and y -> - ∞ as x -> ∞
B. The solution curve y=y(x) is modified exponential growth, and y -> 80 as x -> ∞
C. The solution curve y=y(x) is modified exponential decay, and y -> 80 as x -> ∞
D. The solution curve y=y(x) is modified exponential decay, and y -> -40 as x -> ∞
E. The solution curve y=y(x) is modified exponential growth, and y -> ∞ as x -> ∞
Well, let's take a look at this equation! The differential equation dy/dx = -1.233(y-80) represents the rate of change of y with respect to x.
Now, with an initial condition of y(0)=-40, we can start our journey on the solution curve. Since the right-hand side of the equation involves (y-80), we know that the solution curve will slowly approach but never quite reach y=80.
So, imagine you're hiking up a steep mountain, but you can never actually reach the summit. Similarly, the solution curve will keep getting closer and closer to y=80 but will never actually touch it. But hey, at least you'll have a great view from wherever you end up on that curve!
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.
To solve the given differential equation dy/dx = -1.233(y-80), with initial condition y(0) = -40, we can use the method of separation of variables.
1. Start by separating the variables. We can rewrite the equation as dy / (y - 80) = -1.233 dx.
2. Integrate both sides of the equation. ∫ (1 / (y - 80)) dy = ∫ -1.233 dx.
The integral on the left side can be evaluated as ln|y - 80| + C1, where C1 is the constant of integration.
The integral on the right side is -1.233x + C2, where C2 is another constant of integration.
Therefore, we have ln|y - 80| + C1 = -1.233x + C2.
3. Simplify the equation. We can combine the constants of integration by subtracting C1 from both sides:
ln|y - 80| = -1.233x + (C2 - C1).
4. Take the exponential of both sides. This will remove the natural logarithm:
|y - 80| = e^(-1.233x + (C2 - C1)).
5. Remove the absolute value by considering two cases:
- If y > 80, then y - 80 = e^(-1.233x + (C2 - C1)).
- If y < 80, then -(y - 80) = e^(-1.233x + (C2 - C1)).
6. Solve each case separately:
Case 1: y > 80
y - 80 = e^(-1.233x + (C2 - C1)).
Rearrange the equation to isolate y:
y = 80 + e^(-1.233x + (C2 - C1)).
Case 2: y < 80
-(y - 80) = e^(-1.233x + (C2 - C1)).
Rearrange the equation to isolate y:
y = 80 - e^(-1.233x + (C2 - C1)).
Therefore, the solution curve to the differential equation dy/dx = -1.233(y-80), with initial condition y(0) = -40, is given by either y = 80 + e^(-1.233x + (C2 - C1)) if y > 80, or y = 80 - e^(-1.233x + (C2 - C1)) if y < 80.