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Found in: Page 46

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# In problem ${\mathbf{1}}{\mathbf{-}}{\mathbf{6}}$ , determine whether the differential equation is separable $\frac{\mathbf{dy}}{\mathbf{dx}}{\mathbf{-}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\right)}{\mathbf{=}}{\mathbf{0}}$.

The differential equation $\frac{dy}{dx}-\mathrm{sin}\left(x+y\right)=0$ is not separable.

See the step by step solution

## Step 1: Concept of Separable Differential Equation

A first-order ordinary differential equation $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}{\mathbf{=}}{\mathbit{f}}\left(x,y\right)$ is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions ${\mathbit{g}}\left(x\right)$ that is a function of ${\mathbit{x}}$ alone and ${\mathbit{h}}\left(y\right)$ that is a function of ${\mathbit{y}}$ alone.

Mathematically, the equation is $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}{\mathbf{=}}{\mathbit{f}}\left(x,y\right)$ separable, when ${\mathbit{f}}\left(xy\right){\mathbf{=}}{\mathbit{g}}\left(x\right){\mathbit{h}}\left(y\right)$

## Step 2: Determining whether the equation is Separable or not

The given equation is

$\frac{d}{dx}-\mathrm{sin}\left(x+y\right)=0......\left(1\right)$

Write the given equation as follows:

role="math" localid="1654761564667" $\frac{d}{dx}=\mathrm{sin}\left(x+y\right)......\left(2\right)$

Now, use the trigonometrical to identity $\mathrm{sin}\left(A+B\right)=\mathrm{sin}A\mathrm{cos}B=\mathrm{cos}A\mathrm{sin}B$ . It results,

$\mathrm{sin}\left(x+y\right)=\mathrm{sin}x\mathrm{cos}y+\mathrm{cos}x\mathrm{sin}y$

In this case, equation (2) becomes,

$\frac{dy}{dx}=\mathrm{sin}x\mathrm{cos}y+\mathrm{cos}x\mathrm{sin}y........\left(3\right)$

The function in the right – hand side of equation (3) is

$f\left(x,y\right)=\mathrm{sin}x\mathrm{cos}y+\mathrm{cos}x\mathrm{sin}y$

This function can’t be written as a product of two functions $g\left(x\right)$ and $h\left(y\right)$.

Therefore, the given differential equation is not separable.