Introduction

In this article, we solve a problem involving the slope of a curve at each point and the curve's point of intersection with a given line. This is a common task in differential equations, and understanding how to solve such problems is crucial for many fields, including physics, engineering, and applied mathematics. We will solve the given problem step-by-step, using differential equations to find the equation of the curve.

The Problem

We are given the following information:

The slope of the curve at any point (x, y) is given by the differential equation (frac{dy}{dx} 4xy). The curve passes through the point (0, 4).

The goal is to determine the equation of the curve.

Solving the Problem

We start by analyzing the differential equation (frac{dy}{dx} 4xy). This is a separable differential equation, which means we can separate the variables (y) and (x) on different sides of the equation.

Step 1: Separation of Variables

We rewrite the equation as:

(frac{dy}{y} 4x ; dx)

Step 2: Integration

Next, we integrate both sides:

(int frac{dy}{y} int 4x ; dx)

The left side integrates to:

(ln|y| C_1)

The right side integrates to:

(2x^2 C_2)

We combine these results:

(ln|y| 2x^2 C)

where (C C_2 - C_1) is a constant of integration. To simplify, we exponentiate both sides:

(y e^{2x^2 C})

Let (k e^C), which is a positive constant. Thus, we can write:

(y ke^{2x^2})

Step 3: Applying the Initial Condition

We apply the initial condition given in the problem: when (x 0, y 4). Substituting these values in, we get:

(4 k e^{2(0)^2})

Since (e^{2(0)^2} 1), we find that:

(k 4)

Therefore, the equation of the curve is:

(y 4e^{2x^2})

Verification

To verify our solution, we compute the derivative (y') and check that it matches the given slope:

(y' frac{d}{dx}(4e^{2x^2}) 4 cdot 2 cdot x cdot e^{2x^2} 8x e^{2x^2})

Given the original equation (frac{dy}{dx} 4xy), we substitute:

(y' 4 cdot y cdot x 4 cdot 4e^{2x^2} cdot x 16xe^{2x^2})

This confirms that our solution is correct, as both forms of the derivative are consistent with each other.

Conclusion

In this article, we have solved the problem of finding the equation of a curve given its slope and a point on the curve. The key steps involved separation of variables, integration, solving for the constant of integration using the initial condition, and verification of the solution. Understanding these techniques is essential for solving similar problems in differential equations and calculus.