91Ó°ÊÓ

Differential equations often need to be rewritten in a way that each variable exists on opposite sides of the equation. This process is known as separation of variables.

In our given exercise, we start with the equation:
\( y' = y^2 - e^{3t} y^2 \).
Here, we factor out the common term \( y^2 \) to simplify it to:
\( y' = y^2 (1 - e^{3t}) \).

Next, we aim to separate the variables \( y \) and \( t \). To do this, we rewrite the equation by dividing both sides by \( y^2 (1 - e^{3t}) \) and multiplying both sides by \( dt \). This gives us:
\( \frac{dy}{y^2(1 - e^{3t})} = dt \).

Now, each side of the equation has only one variable, prepared for the next step.

Integration is a method to find functions from their derivatives, solving for the original function.

After we separated our variables, we must integrate both sides of the equation. Integrating the left side:
\( \text{LHS} = \frac{1}{y^2} \bigg/ (1 - e^{3t}) \bigg) dy \),
and the right side: \( \text{RHS} = dt \),
results in:
\( \frac{1}{y^2} dy = \bigg[ t - \frac{e^{3t}}{3} \bigg] dt + C \).
Here, \( C \) is the constant of integration added as a result of indefinite integration.

Integrating \( \frac{1}{y^2} dy \) gives us:
\( -\frac{1}{y} \), and integrating \( (1 - e^{3t}) dt \) results in: \( t - \frac{e^{3t}}{3} \). Hence, our equation becomes:
\( -\frac{1}{y} = t - \frac{e^{3t}}{3} + C \).

Initial conditions are specific values that help us determine the particular solution when solving a differential equation.

In this exercise, we are given the initial condition: \( y(0) = 1 \). This means when \( t = 0 \), \( y = 1 \).
We use this to find \( C \), the constant of integration.

We substitute \( t = 0 \) and \( y = 1 \):
\( -\frac{1}{1} = 0 - \frac{e^{0}}{3} + C \).
Simplifying this, since \( e^{0} = 1 \), we get:
\( -1 = 0 - \frac{1}{3} + C \).
Solving for \( C \) yields:
\( -1 = -\frac{1}{3} + C \),
\( C = -1 + \frac{1}{3} \),
\( C = -\frac{2}{3} \).

Thus, we have determined our constant of integration.

The final step after determining our integration constant is to solve for the variable \( y \). Using our previous results and the determined constant, we rearrange our equation.

We substitute \( C = -\frac{2}{3} \) into the equation:
\( -\frac{1}{y} = t - \frac{e^{3t}}{3} - \frac{2}{3} \).
To isolate \( y \), we multiply both sides by -1:
\( \frac{1}{y} = -t + \frac{e^{3t}}{3} + \frac{2}{3} \).
Finally, solving for \( y \),
\( y = \frac{1}{-t + \frac{e^{3t}}{3} + \frac{2}{3}} \).
Therefore, the solution to the differential equation using separation of variables, integration, and applying initial conditions is:
\( y = \frac{1}{ -t + \frac{e^{3t}}{3} + \frac{2}{3}} \).