Solving and Understanding the Differential Equation dy/dx xy^4

Introduction

The differential equation (frac{dy}{dx} xy^4) is a separable first-order ordinary differential equation (ODE). Such equations can be solved by separating the variables, which allows us to integrate each variable separately to find the solution. This article will guide you through the process of solving this specific ODE and provide insights into its solution.

Solution Method

To solve the equation (frac{dy}{dx} xy^4), we start by separating the variables (y) and (x). This is done by dividing both sides by (y^4) and multiplying both sides by (dx). This gives us:

[frac{dy}{y^4} x , dx]

Integration

Next, we integrate both sides of the equation:

[int frac{dy}{y^4} int x , dx]

The left side integrates to:

[int frac{dy}{y^4} int y^{-4} , dy frac{y^{-3}}{-3} C_1 -frac{1}{3y^3} C_1]

The right side integrates to:

[int x , dx frac{x^2}{2} C_2]

Setting these equal gives us:

[-frac{1}{3y^3} C_1 frac{x^2}{2} C_2]

Combining the constants (C_1) and (C_2) into a single constant (C), we get:

[-frac{1}{3y^3} frac{x^2}{2} C]

Solving for y

To solve for (y), we first isolate the expression involving (y):

[frac{1}{3y^3} -frac{x^2}{2} - C]

Multiplying both sides by (-3), we obtain:

[y^3 -frac{3}{2x^2} - 3C]

Introducing a new constant (C'), we have:

[y^3 -frac{3}{2x^2} - 3C']

Finally, taking the cube root of both sides, we get the solution:

[y sqrt[3]{-frac{3}{2x^2} - 3C'}]

Note on the Solution

It's important to note a few things about the solution: The solution involves a cube root, so (y) may have multiple real or complex values depending on the values of (x) and (C'). The equation (frac{dy}{dx} xy^4) is only valid for (x eq 0) and ( y eq -frac{1}{sqrt[4]{x}}) to avoid division by zero or undefined terms. Geometrically, the general solution represents a family of curves. For (C 0), the curve is a line shifted by a constant value. Other particular solutions can be found by imposing initial conditions, leading to specific Cauchy problems.

Conclusion

In conclusion, solving the differential equation (frac{dy}{dx} xy^4) involves separating variables and integrating the resulting expressions. The solution is a family of curves that depend on the constant (C). Understanding this solution requires careful interpretation of the constants and the variables involved.

Keywords: differential equation, separable variable, calculus

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