Chapter 1: Q10 E (page 14)

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.
$y - \log_{e} y = x^{2} + 1$ , $\frac{dy}{dx} = \frac{2 xy}{y - 1}$

Short Answer

Expert verified

The given relation is an implicit solution to the given differential equation.

Step by step solution

Differentiating the given relation

As, in the given relation $y - \log_{e} y = x^{2} + 1$ , y is defined implicitly as the function of x, so by using implicit differentiation, we will differentiate the given relation concerning x,

$\frac{d}{dx} (y - \log_{e} y) = \frac{d}{dx} (x^{2} + 1) \frac{dy}{dx} - \frac{1}{y} \frac{dy}{dx} = 2 x$

Simplification of the differential equation obtained in step 1

$(1 - \frac{1}{y}) \frac{dy}{dx} = 2 x (\frac{y - 1}{y}) \frac{dy}{dx} = 2 x \frac{dy}{dx} = \frac{2 xy}{y - 1}$

Which is identical to the given differential equation.

Thus, the relation $y - \log_{e} y = x^{2} + 1$ is an implicit solution to the differential equation $\frac{dy}{dx} = \frac{2 xy}{y - 1}$ .

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In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.
$y - \log_{e} y = x^{2} + 1$ , $\frac{dy}{dx} = \frac{2 xy}{y - 1}$

Short Answer

Step by step solution

Differentiating the given relation

Simplification of the differential equation obtained in step 1

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In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.y-logey=x2+1,dydx=2xyy-1

Short Answer

Step by step solution

Differentiating the given relation

Simplification of the differential equation obtained in step 1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Logic and Functions

Discrete Mathematics

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.
$y - \log_{e} y = x^{2} + 1$ , $\frac{dy}{dx} = \frac{2 xy}{y - 1}$