A curve is such that dy/dx = - k/x^2, where k is constant. Given that curve passes through (6,2.5) a
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Published on May 23, 2025
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Table of Contents
Introduction
In this tutorial, we will solve a differential equation representing a curve defined by the relationship dy/dx = -k/x², where k is a constant. We will also determine the specific curve that passes through the point (6, 2.5). This tutorial is applicable for students of IGCSE AS & A level mathematics and provides a systematic approach to solving such problems through integration.
Step 1: Understand the Differential Equation
- The given differential equation is [ \frac{dy}{dx} = -\frac{k}{x^2} ]
- This represents the rate of change of y with respect to x. The negative sign indicates that as x increases, y decreases.
Step 2: Separate Variables
- To solve the differential equation, we will separate the variables y and x [ dy = -\frac{k}{x^2} dx ]
- This allows us to integrate both sides independently.
Step 3: Integrate Both Sides
- Integrate the left side with respect to y and the right side with respect to x [ \int dy = \int -\frac{k}{x^2} dx ]
- The left side integrates to y [ y = \int -\frac{k}{x^2} dx ]
- The right side can be solved as follows [ \int -\frac{k}{x^2} dx = k \cdot \frac{1}{x} + C ]
- Thus, we have [ y = \frac{k}{x} + C ]
- Where C is the constant of integration.
Step 4: Apply the Initial Condition
- Since the curve passes through the point (6, 2.5), we will use this point to find the constants k and C.
- Substitute x = 6 and y = 2.5 into the equation [ 2.5 = \frac{k}{6} + C ]
- This gives us our first equation.
Step 5: Express C in Terms of k
- Rearranging the equation gives [ C = 2.5 - \frac{k}{6} ]
- We will retain this expression for C for later use.
Step 6: Determine k
- To find k, we need another condition. Since k is a constant, it can be defined by additional conditions or characteristics of the curve. If none are given, use any reasonable assumption based on context or additional data.
Step 7: Final Equation of the Curve
- Substitute the expression for C back into the equation of the curve [ y = \frac{k}{x} + \left(2.5 - \frac{k}{6}\right) ]
- This simplifies to [ y = \frac{k}{x} + 2.5 - \frac{k}{6} ]
Conclusion
In summary, we have derived the equation of the curve given the differential equation and initial condition. The final form of the curve, dependent on k, is
[ y = \frac{k}{x} + 2.5 - \frac{k}{6} ] To proceed, you would typically need additional information to determine the specific value of k. This method illustrates the process of solving differential equations through integration and applying initial conditions effectively.