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Chapter 43: Problem 1110

Find an equation of a curve, the slope of which is equal to \(2 \mathrm{x}\) at any point on the curve.

Short Answer

Expert verified
The equation of the curve whose slope is equal to 2x at any point is \(y = x^2 + C\), where C is an arbitrary constant.

Step by step solution

01

Write down the given differential equation

We are given that the slope of the curve is equal to 2x. In other words, the derivative of the curve with respect to x is 2x. We can write this as a differential equation: \(\frac{dy}{dx} = 2x\)
02

Integrate the differential equation

Integrate both sides of the differential equation with respect to x: \[\int \frac{dy}{dx} dx = \int 2x dx\] By integrating both sides, we get: \[y = x^2 + C\] where C is the constant of integration.
03

Write down the final equation

The equation of the curve whose slope is equal to 2x at any point is as follows: \[y = x^2 + C\] with C being an arbitrary constant.

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