Q9 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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.

${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{4}$, $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{x}}{\mathbf{y}}$

Step-by-Step Solution

Answer

The given relation is not an implicit solution to the given differential equation.

Step1: Differentiating the given relation

As, in the given relation ${\mathrm{x}}^2+{\mathrm{y}}^2=4$, y is defined implicitly as the function of x, so by using implicit differentiation, we will differentiate the given relation concerning x,

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)=\frac{\mathrm{d}}{\mathrm{dx}}\left(4\right) \\ 2\mathrm{x}+2\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}=0 \end{gathered}$$

Step 2: Simplification of the differential equation obtained in step 1

$$\begin{gathered} 2\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}=-2\mathrm{x} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{2\mathrm{x}}{2\mathrm{y}} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{x}}{\mathrm{y}} \end{gathered}$$

Which is not identical to the given differential equation.

Thus, the relation ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{4}$ is not an implicit solution to the differential equation $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{x}}{\mathbf{y}}$.