Understanding and Solving the Differential Equation y' 3/x^3

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The given differential equation y' 3/x^3 presents a unique challenge that is commonly encountered in calculus, specifically in the area of differential equations. This equation is not immediately straightforward and requires the application of certain techniques to find its solution. In this article, we will explore how to solve this type of differential equation, discussing the process and providing a step-by-step solution.

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What is a Differential Equation?

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A differential equation is a mathematical equation that relates a function with one or more of its derivatives. In other words, it describes the relationship between a quantity and its instantaneous rate of change. These equations are fundamental in many areas of science and engineering, and they are used to model a wide range of phenomena including physical systems, population dynamics, and economic forecasts.

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Solving the Given Differential Equation

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The differential equation given is:

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y' 3/x^3

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Let's solve it step-by-step.

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Step 1: Separation of Variables

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The first step in solving this differential equation is to separate the variables. We can rewrite the equation as:

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dy 3/x^3 dx

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This step involves moving all terms involving y on one side and all terms involving x on the other side of the equation.

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Step 2: Integration

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Next, we integrate both sides of the equation:

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∫ dy ∫ 3/x^3 dx

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The integral on the left side is straightforward:

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∫ dy y

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For the right side, we need to integrate:

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∫ 3/x^3 dx 3 ∫ x^(-3) dx

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We know that the integral of x^n is (x^(n 1))/(n 1), for n ≠ -1. Therefore:

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3 ∫ x^(-3) dx 3 * (-1/2) * x^(-2) -3/2 * x^(-2)

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Adding the constant of integration C, the equation becomes:

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y -3/2 * x^(-2) C

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Which can be rewritten as:

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y -3/(2x^2) C

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Interpreting the Solution

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The general solution to the differential equation y' 3/x^3 is y -3/(2x^2) C, where C is an arbitrary constant. This constant can be determined if an initial condition is given, such as y(x0) y0, where x0 and y0 are specific values.

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Conclusion

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In summary, the process of solving the differential equation y' 3/x^3 involves separating variables, integrating both sides, and then interpreting the general solution based on the initial condition, if provided.