91Ó°ÊÓ

One of the most common methods to solve a first-order differential equation is separation of variables. This technique is useful when the derivatives of two variables can be expressed as separate functions on different sides of an equation. For instance, in the differential equation \( \frac{d y}{d x} = 3x^2 e^{-y} \), the goal is to manipulate it so that all terms involving \( y \) are on one side and all terms involving \( x \) are on the other.

To begin, identify which part of the equation relates to \( y \) and which to \( x \). In our example, \( e^{-y} \) involves \( y \), and \( 3x^2 \) involves \( x \). We rearrange the equation so that these terms are isolated with their respective differentials, resulting in \( e^y \, dy = 3x^2 \, dx \).

Now the equation is expressed with each variable on opposite sides. By doing this, we've separated the variables, paving the way to solve the equation by integration.

Once we have separated the variables in our differential equation, the next step is integration. Integrating both sides of the equation helps us find a relation between the variables involved, typically leading towards a more straightforward expression of the solution.

For our differential equation \( e^y \, dy = 3x^2 \, dx \), we integrate each side with respect to its corresponding variable:

• Integrate the left side, \( \int e^y \, dy \), to get \( e^y + C_1 \).
• Integrate the right side, \( \int 3x^2 \, dx \), to find \( x^3 + C_2 \).

Combining these results, we form \( e^y = x^3 + C_2 - C_1 \). To simplify, let constant \( C = C_2 - C_1 \), thus giving us \( e^y = x^3 + C \).

Integration is crucial as it transitions us from a rate of change (differential form) to an expression that relates the quantities directly.

After integration, our goal is often to determine the general solution of the differential equation. The general solution encompasses a family of solutions that fit the overall behavior dictated by the equation.

For the differential equation \( e^y = x^3 + C \), obtained post-integration, we need to express \( y \) explicitly in terms of \( x \). This requires solving for \( y \):

• Apply the natural logarithm to both sides of the equation: \( y = \ln(x^3 + C) \).

This expression, \( y = \ln(x^3 + C) \), is the general solution. It indicates that for every different value of the constant \( C \), there is a specific solution curve on a graph.

Understanding the general solution allows us to see all possible solutions to the differential equation, making it a powerful tool in mathematical problem-solving.