What is the Equation of a Curve with a Given Slope and Passing Through a Point?

This article delves into the mathematical process of finding the equation of a curve whose slope is given in terms of the coordinates and which passes through a specific point. This problem involves the application of differential equations and integration techniques.

The Problem Statement

The initial problem statement is to find the equation of a curve whose slope, defined as the derivative of y with respect to x, is given by y^2/x. Additionally, the curve must pass through the point (4, 14).

Solution Method

We start with the given differential equation:

dy/dx y^2/x

This is a separable differential equation. To solve it, we separate the variables y and x on different sides:

dy/y^2 dx/x

Integrating both sides, we get:

-1/y ln|x| - C

Rearranging, we find:

1/y C - ln|x|

Now, to find the constant C, we use the initial condition xy 14 at the point (4, 14):

1/14 C - ln|4|

Solving for C:

C 1/14 ln|4|

Substituting this value of C back into the equation:

1/y 1/14 ln|4| - ln|x|

Combining the logarithmic terms:

1/y 1/14 ln|4/|x|

Further simplifying:

y 14 / (1 14 * ln|4/|x|

Since we need the equation in a more compact form, we can rearrange it to:

y 14 / (1 - 4 * ln|x|)

Verification via Graphical Means

To verify our work, we can plot the curve using y 14 / (1 - 4 * ln|x|) and check if it passes through the point (4, 14) and if the slope matches the given condition. This graphical verification helps in ensuring the correctness of the derived equation.

Conclusion

In summary, we have derived the equation of the curve whose slope at any point (x, y) is given by y^2/x and which passes through the point (4, 14). The final equation is:

y 14 / (1 - 4 * ln|x|)

This solution involves the use of differential equations and integration techniques, providing a practical example of solving problems in calculus.