91Ó°ÊÓ

First-order differential equations involve the first derivative of a function. These equations can often describe various real-world phenomena, such as population growth or the rate of cooling of an object. In a first-order differential equation, you will typically see the notation \(\frac{dy}{dx}\) or \(y'\). The general form is:
\[ \frac{dy}{dx} = f(x, y) \]
Here, \(y'\) means the rate of change of \(y\) with respect to \(x\). In the exercise, the first-order differential equation given is: \[ y' = 5y^{-2} \]
This tells us how \(y\) changes as \(x\) changes, and we need to follow a method to find the function \(y\) itself. With practice, solving these becomes a more intuitive task. Remember, the goal is to find \(y\) as a function of \(x\).

Separation of variables is a powerful method used to solve first-order differential equations. This technique involves rearranging the equation so that each variable and its differential are on opposite sides of the equation. Here is how we handle the exercise step-by-step:
1. Start with the given equation: \[ y' = 5y^{-2} \]
2. Rewrite \(y'\) as \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = 5y^{-2} \]
3. Separate the variables by multiplying both sides by \(dx\) and \(y^2\): \[ y^2 dy = 5 dx \]
4. Integrate both sides to find \(y\): \[ \int y^2 dy = \int 5 dx \]
This results in: \[ \frac{y^3}{3} = 5x + C \]
Where \(C\) is the constant of integration.

Initial conditions are vital to find the specific solution of a differential equation. They give us the starting point, which allows us to solve for the constant of integration. In the exercise, the initial condition provided is \(y=3\) when \(x=2\).
Let's apply this initial condition: \[ \frac{y^3}{3} = 5x + C \]
Substitute \(y=3\) and \(x=2\): \[ \frac{3^3}{3} = 5(2) + C \]
Which simplifies to: \[ 9 = 10 + C \]
Thus, \(C = -1\).
We can now write the specific solution as: \[ \frac{y^3}{3} = 5x - 1 \]
Solving for \(y\), we find: \[ y^3 = 15x - 3 \]
Finally, \[ y = \sqrt[3]{15x - 3} \]
This is the solution that satisfies both the differential equation and the initial conditions.