Question

The differential equation of the family of curves, ${\mathrm{x}}^2=4\mathrm{b}\left(\mathrm{y}+\mathrm{b}\right),\mathrm{b}\in \mathrm{R}$, is:

• A. $\mathrm{xy}\text{'}\text{'}=\mathrm{y}\text{'}$
• B. $\mathrm{x}{\left(\mathrm{y}\text{'}\right)}^2=\mathrm{x}+2\mathrm{yy}\text{'}$
• C. $\mathrm{x}{\left(\mathrm{y}\text{'}\right)}^2=\mathrm{x}-2\mathrm{yy}\text{'}$
• D. ${\mathrm{x}}^2=2\mathrm{yy}\text{'}-\mathrm{x}$

Answer 1

The correct option is B

$\mathrm{x}{\left(\mathrm{y}\text{'}\right)}^2=\mathrm{x}+2\mathrm{yy}\text{'}$

Explanation for the correct option:

Given, ${\mathrm{x}}^2=4\mathrm{b}\left(\mathrm{y}+\mathrm{b}\right)$

On differentiating, we get;

$$\begin{gathered} 2\mathrm{x}=4\mathrm{b}\frac{\mathrm{dy}}{\mathrm{dx}} \\ \mathrm{b}=\frac{\mathrm{x}}{2\left({\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right)} \end{gathered}$$

On substituting the value of $\mathrm{b}$in the given equation, we get;

$$\begin{gathered} {\mathrm{x}}^2=4\left(\frac{\mathrm{x}}{2\left({\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right)}\right)\left(\mathrm{y}+\left(\frac{\mathrm{x}}{2\left({\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right)}\right)\right) \\ \mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2=2\left(2\mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+\mathrm{x}\right) \\ \mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2=2\mathrm{x}+4\mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) \end{gathered}$$

Hence, the correct answer is option (B).