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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 role="math" localid="1654775979001" $\mathbf{\left(}\mathbit{x}{\mathbit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{3}{\mathbit{y}}^{\mathbf{2}}\mathbf{\right)}\mathbit{d}\mathbit{y}\mathbf{-}\mathbf{2}\mathbit{x}\mathbf{}\mathbit{d}\mathbit{x}\mathbf{=}\mathbf{0}$.

The differential equation $\left(x{y}^{2}+3{y}^{2}\right)dy-2xdx=0$ is 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 alone.

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

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

The given equation is

$\left(x{y}^{2}+3{y}^{2}\right)dy-2xdx=0..........\left(1\right)$

Equation (i) can be written as

${y}^{2}\left(x+3\right)dy-2xdx=0\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}=\frac{2x}{{y}^{2}\left(x+3\right)}$

The function in the right – hand side of equation (1) is

$f\left(x,y\right)=\frac{2x}{{y}^{2}\left(x+3\right)}\phantom{\rule{0ex}{0ex}}=\frac{2x}{{x}^{2}+3}\frac{1}{{y}^{2}}$

This function can be written as a product of two functions $g\left(x\right)$ and $h\left(y\right)$defined as,

$g\left(x\right)=\frac{2x}{{x}^{2}+3}\phantom{\rule{0ex}{0ex}}=\frac{1}{{y}^{2}}$

Therefore, the given differential equation is separable.