A curve is such that dy/dx = - k/x^2, where k is constant. Given that curve passes through (6,2.5) a

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.