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Chapter 43: Problem 1114

Find a curve having its slope always equal to half the abscissa, and passing through \((0,-3)\).

Short Answer

Expert verified
The curve having its slope always equal to half the abscissa and passing through $(0,-3)$ is given by the equation \( y = \frac{1}{4}x^2 - 3 \).

Step by step solution

01

Write down the differential equation

The problem provides us with the slope of the curve at any point \( x \), which defines a first order differential equation. The slope of the curve - which is the derivative of \( y \) with respect to \( x \), or \( y' \), is given as half of the abscissa (or \( x \) coordinate). This can be written as the following differential equation: \( y' = \frac{1}{2}x \).
02

Solve the differential equation

This differential equation is a simple first order differential equation and can be solved by integrating both sides with respect to \( x \). When you integrate the left side, you get \( y \), and when you integrate the right side, you get \(\frac{1}{4}x^2 \). So, \[ y = \frac{1}{4}x^2 + C \], where \( C \) is an unknown constant we will determine using the provided point on the curve.
03

Find the constant C using the provided point

Now we need to find the constant \( C \) which yields the particular solution to this differential equation. We were given that the curve passes through the point (0,-3). Let's substitute these values into our equation. Substituting \( x = 0 \) and \( y = -3 \), we have \[ -3 = \frac{1}{4} * 0^2 + C \] Solving for \( C \) gives \( C = -3 \).
04

Write down the final equation of the curve

This yields the final equation of the curve: \[ y = \frac{1}{4}x^2 - 3 \] Therefore, the unknown curve that always has a slope equals to half the abscissa and passes through the point (0,-3) is given by \( y = \frac{1}{4}x^2 - 3 \).

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Most popular questions from this chapter

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