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Question

# The solution of the differential equation $9y\left(\frac{dy}{dx}\right)+4x=0$ is

A

$\left(\frac{{y}^{2}}{9}\right)+\left(\frac{{x}^{2}}{4}\right)=c$

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B

$\left(\frac{{y}^{2}}{4}\right)+\left(\frac{{x}^{2}}{9}\right)=c$

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C

$\left(\frac{{y}^{2}}{9}\right)-\left(\frac{{x}^{2}}{4}\right)=c$

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D

${\left(y\right)}^{2}+\left(\frac{{x}^{2}}{9}\right)=c$

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Solution

## The correct option is B $\left(\frac{{y}^{2}}{4}\right)+\left(\frac{{x}^{2}}{9}\right)=c$Explanation for correct optionGiven: $9y\left(\frac{dy}{dx}\right)+4x=0$$⇒9y\left(\frac{dy}{dx}\right)=-4x\phantom{\rule{0ex}{0ex}}⇒9y·dy=-4x·dx$integrating both sides$⇒\int 9y·dy=-\int 4x·dx\phantom{\rule{0ex}{0ex}}⇒9\frac{{y}^{2}}{2}+{c}_{1}=-\frac{4{x}^{2}}{2}+{c}_{2}\phantom{\rule{0ex}{0ex}}⇒\frac{9{y}^{2}}{2}+\frac{4{x}^{2}}{2}={c}_{2}$dividing both sides by $36$$⇒\frac{9{y}^{2}}{36}+\frac{4{x}^{2}}{36}=\frac{2}{36}{c}_{2}\phantom{\rule{0ex}{0ex}}⇒\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=c$Hence, option B is correct.

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