91Ó°ÊÓ

In solving differential equations, one effective method is the separation of variables.
This technique is helpful for equations that can be written such that all terms involving the dependent variable are on one side,
and all terms involving the independent variable are on the other.
For example, consider the differential equation: \[ y^{\text{'}}=\frac{e^{2t}}{y^{2}} \].
Here, we can rewrite this as \[ y^{2} dy = e^{2t} dt \].
By rearranging terms, you get all the terms in \( y \) on one side (\( y^2 dy \)), and all the terms in \( t \) on the other side (\( e^{2t} dt \)).
This simplification makes the equation easier to solve.

Integration is a fundamental concept used to find solutions to differential equations.
Once you have separated the variables, the next step is to integrate both sides.
With the separated form \[ y^{2} dy = e^{2t} dt \],
you integrate each side with respect to its own variable: \[ \int y^{2} dy = \int e^{2t} dt \].
The integration yields \[ \frac{1}{3} y^{3} = \frac{1}{2} e^{2t} + C \].
This result gives us a relation between \( y \) and \( t \), with \( C \) being an integration constant.
Integration can often involve standard formulas or techniques such as substitution and partial fractions.
Be comfortable recognizing and applying these to simplify the process.

When performing integration, a constant of integration appears due to the antiderivative uncertainness.
This constant accounts for all indefinite integrals and ensures the solution's generality.
For example, \[ \frac{1}{3} y^{3} = \frac{1}{2} e^{2t} + C \],
where \( C \) is the constant of integration.
Every indefinite integral will include such a constant to represent an entire family of antiderivatives.
While solving for the dependent variable, the integration constant may take on different forms but essentially represents the same idea.
In the next step, we solve for \( y \) and obtain \[ y = \sqrt[3]{\frac{3}{2} e^{2t} + C_1} \],
where \( C_1 \) is just another representation of \( C \).
Note that constants can be adjusted during algebraic manipulation but always signify an inherent part of the solution.