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Magazine Chapter 3: Problem 16
Find the \(x y\) -equation of the curve through (1,2) whose slope at any point is three times the square of its \(y\) -coordinate.
Short Answer
Expert verified
The equation of the curve is \( y = \frac{1}{-3x + \frac{7}{2}} \).
Step by step solution
01
Understand the Problem
We are asked to find the equation of a curve. This curve has the property that its slope at any point \((x, y)\) is given by \(\frac{dy}{dx} = 3y^2\). We also know that the curve passes through the point \((1, 2)\).
02
Set Up the Differential Equation
Given that the slope \(\frac{dy}{dx} = 3y^2\), we have a differential equation to solve: \[\frac{dy}{dx} = 3y^2\].
03
Separation of Variables
Separate variables to prepare for integration. \[\frac{dy}{y^2} = 3\, dx\]This separates the variables \(y\) and \(x\) on different sides of the equation.
04
Integrate Both Sides
Integrate both sides of the equation:\[\int \frac{dy}{y^2} = \int 3\, dx\]The left side integrates to \(-\frac{1}{y}\) and the right side to \(3x + C\), where \(C\) is the integration constant.
05
Solve for the Integration Constant
Equating the integrated equation gives:\[-\frac{1}{y} = 3x + C\]Using the point \((1, 2)\), substitute \(x = 1\) and \(y = 2\):\[-\frac{1}{2} = 3(1) + C\]Solving for \(C\) gives \(C = -\frac{7}{2}\).
06
Write the General Equation
Substitute back \(C = -\frac{7}{2}\) into the integrated equation: \[-\frac{1}{y} = 3x - \frac{7}{2}\]Rearranging gives:\[\frac{1}{y} = -3x + \frac{7}{2}\]
07
Solve for y in terms of x
Rearrange the equation \(\frac{1}{y} = -3x + \frac{7}{2}\) to solve for \(y\):\[y = \frac{1}{-3x + \frac{7}{2}}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope of a Curve
The slope of a curve at a given point indicates how steep the curve is at that specific location. In our problem, the slope is given by the derivative \( \frac{dy}{dx} = 3y^2 \), which means it depends on the square of the \( y \)-coordinate at any point. This tells us that:
- If \( y \) increases, the slope becomes steeper quickly because of the square term.
- The slope is always non-negative because it is based on \( y^2 \), which is positive for all real numbers.
Separation of Variables
Separation of variables is a technique used to solve differential equations. It involves rearranging the equation so that each variable and its derivative are on opposite sides of the equation. In our problem:\[ \frac{dy}{dx} = 3y^2 \]is rewritten as:\[ \frac{dy}{y^2} = 3 \, dx \]This step is essential because it allows us to integrate both sides independently. By separating the variables, you can integrate them with respect to their own independent variables, enabling a solution to be found for each part.
Integration Constant
When we integrate both sides, we introduce an integration constant \( C \). The integration process is:\[ \int \frac{dy}{y^2} = \int 3 \, dx \]which leads to:\[ -\frac{1}{y} = 3x + C \]This constant represents an unknown value that arises because indefinite integration can result in multiple solutions that differ by a constant. To find the specific value of \( C \), we use a known point on the curve鈥攊n this exercise, the point \((1, 2)\). By substituting \( x = 1 \) and \( y = 2 \), we can solve for \( C \), pinning down the particular solution that satisfies the initial condition.
Curve Through a Point
Determining a curve through a specific point ensures that the equation we derive passes exactly through it. After calculating the integral and obtaining an equation like \[ -\frac{1}{y} = 3x + C \], substituting the given point, \( (1, 2) \), helps:
- Verify the correctness of the equation.
- Find the specific integration constant \( C \), ensuring the solution is uniquely tied to the initial condition.
