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Separation of variables is a powerful technique used in solving differential equations. It works particularly well with equations that can be rearranged so that all terms involving one variable appear on one side of the equation, while all terms involving the other variable appear on the opposite side. This method is akin to sorting out different fruits into separate baskets, allowing you to handle each type independently.

In the given differential equation, \(\frac{dy}{dx} = \frac{ky}{x}\), our goal is to rearrange the terms to isolate the variables. By concentrating all \(y\) terms on one side and all \(x\) terms on the other, we achieve separation.

We can rewrite the equation as \(\frac{dy}{y} = \frac{k}{x}dx\). This spells out that while \(x\) affects the right-hand side, the left is solely impacted by \(y\). This neat separation paves the path for the next step: integration.

Integration comes into play once the variables are separated. It is like summing up small intervals to find the entire area under the curve, or more formally, finding the antiderivative. This step is crucial in finding an equation that expresses the relationship between \(y\) and \(x\).

With the separated equation \(\frac{dy}{y} = \frac{k}{x} dx\), we integrate both sides independently. The left-hand side becomes \(\int \frac{1}{y} \, dy = \ln|y|\), while the right-hand side becomes \(\int \frac{k}{x} \, dx = k \ln|x|\).

• The natural logarithm \(\ln\) emerges because it is the antiderivative of the \(\frac{1}{u}\) function.
• It is essential to add \(C\), the constant of integration, at this stage to reflect all possible solutions resulting from indefinite integration.

Together, these integrals give us the equation \(\ln|y| = k \ln|x| + C\).

The general solution provides an expression that describes all possible solutions of the differential equation. It's like having a blueprint that covers every conceivable structure built within given parameters.

To solve \(\ln|y| = k \ln|x| + C\), you exponentiate both sides to eliminate the natural logarithms, resulting in \(|y| = e^{k \ln|x| + C}\). This turns the equation into a more manageable form, illustrating the relationship in a clearer way.

You can further simplify this to \(|y| = |x|^k e^C\). Letting \(C_1 = e^C\) simplifies even more: observe that because of the properties of exponents, you get \(y = C_1 x^k\). Here, \(C_1\) is a constant that can be adjusted to either positive or negative, offering a full spectrum of solutions.

• This is called the general solution because it encompasses all the particular solutions for any initial condition.
• This form, \(y = C x^k\), is the fundamental answer reflecting how \(y\) varies with \(x\) given the parameters \(k\) and \(C\).