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Differential equations are mathematical expressions that relate a function with its derivatives. They are essential in modeling numerous real-world phenomena, such as population growth, heat conduction, and fluid dynamics. In the given exercise, we have a first-order differential equation: \( \frac{\mathrm{d} y}{\mathrm{~d} x} = 2 x y \). Here, the rate of change of \( y \) with respect to \( x \) is proportional to the product of \( x \) and \( y \). This can describe processes where growth accelerates at higher values of \( x \). One common approach to solve such equations is the _method of separation of variables_. This technique involves rearranging the equation to isolate different variables on opposite sides, allowing for integration. This method is effective in situations where the equation can be decomposed, but it may not work for every form of differential equation. The primary idea is to express all \( y \)-dependent terms on one side and all \( x \)-dependent terms on the other, as shown in the steps of the solution.

The power series method is another powerful tool for solving differential equations. This approach involves assuming that the solution can be represented as a power series, which is an infinite sum of terms with increasing powers of \( x \). For the problem at hand, we start with the series expression: \( y = \sum_{m=0}^{\infty} a_{m} x^{m} \). This method is particularly useful when the solution is expected to be analytic, meaning it has a power series expansion that converges within some interval. By differentiating the series term-by-term, and then substituting back into the differential equation, we establish a relationship between the series coefficients. This relationship helps compute each coefficient systematically.The challenge lies in equating and aligning terms of the same order, usually requiring an index shift. Ultimately, the power series method can transform complex differential equations into a sequence of algebraic problems, making them easier to solve.

Integral calculus plays a vital role in solving differential equations, especially when using the method of separation of variables. In the given exercise, once the equation \( \frac{1}{y} \mathrm{d}y = 2x \mathrm{d}x \) is separated, integration comes next. Integration allows us to find an antiderivative, which, in essence, "undoes" differentiation. When integrating \( \frac{1}{y} \), we obtain \( \ln |y| \), a common result when integrating a function inverse. On the other hand, integrating \( 2x \) yields \( x^2 + C \), where \( C \) is the constant of integration representing any constant term that might be added to the solution.By solving the integrals on both sides, we can derive expressions for \( y \) and \( x \) that satisfy the initial differential equation. Exponentiating the result, we eliminate the natural logarithm, leading to solutions in terms of exponentials, which frequently occur in these contexts.