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Q-5E

Expert-verifiedFound in: Page 46

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" $\mathbf{(}\mathit{x}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{3}{\mathit{y}}^{\mathbf{2}}\mathbf{)}\mathit{d}\mathit{y}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{}\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$.**

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

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

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

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.**

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