Question

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

• A. $\left(\frac{y^2}{9}\right)+\left(\frac{x^2}{4}\right)=c$
• B. $\left(\frac{y^2}{4}\right)+\left(\frac{x^2}{9}\right)=c$
• C. $\left(\frac{y^2}{9}\right)-\left(\frac{x^2}{4}\right)=c$
• D. ${\left(y\right)}^2+\left(\frac{x^2}{9}\right)=c$

Answer 1

The correct option is B

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

Explanation for correct option

Given: $9y\left(\frac{dy}{dx}\right)+4x=0$

$$\begin{gathered} \Rightarrow 9y\left(\frac{dy}{dx}\right)=-4x \\ \Rightarrow 9y·dy=-4x·dx \end{gathered}$$

integrating both sides

$$\begin{gathered} \Rightarrow \int 9y·dy=-\int 4x·dx \\ \Rightarrow 9\frac{y^2}{2}+c_1=-\frac{4x^2}{2}+c_2 \\ \Rightarrow \frac{9y^2}{2}+\frac{4x^2}{2}=c_2 \end{gathered}$$

dividing both sides by $36$

$$\begin{gathered} \Rightarrow \frac{9y^2}{36}+\frac{4x^2}{36}=\frac{2}{36}{c}_2 \\ \Rightarrow \frac{x^2}{9}+\frac{y^2}{4}=c \end{gathered}$$

Hence, option B is correct.