Key Concepts and Formulas

• Differential Equation Formation: The process involves eliminating arbitrary constants from a given equation by differentiation. The number of times you differentiate should match the number of arbitrary constants.
• Differentiation: Applying the rules of calculus to find the derivative of a function.
• Algebraic Manipulation: Rearranging equations to isolate variables or simplify expressions.

Step-by-Step Solution

Step 1: Write the given equation.

The given equation is: $x^2 = 4b(y + b)$

Step 2: Differentiate both sides with respect to x.

We differentiate both sides of the equation with respect to $x$ to introduce $y'$ (which is $\frac{dy}{dx}$). This is necessary to eliminate the arbitrary constant $b$. $\frac{d}{dx}(x^2) = \frac{d}{dx}[4b(y + b)]$ $2x = 4b\frac{d}{dx}(y + b)$ $2x = 4b\frac{dy}{dx}$ $2x = 4by'$ $x = 2by'$

Step 3: Solve for b.

We now solve for $b$ in terms of $x$ and $y'$: $b = \frac{x}{2y'}$

Step 4: Substitute the value of b back into the original equation.

Substitute the expression for $b$ back into the original equation $x^2 = 4b(y + b)$: $x^2 = 4\left(\frac{x}{2y'}\right)\left(y + \frac{x}{2y'}\right)$ $x^2 = \frac{2x}{y'}\left(y + \frac{x}{2y'}\right)$

Step 5: Simplify the equation.

Divide both sides by $x$ (assuming $x \neq 0$): $x = \frac{2}{y'}\left(y + \frac{x}{2y'}\right)$ Multiply both sides by $y'$: $xy' = 2\left(y + \frac{x}{2y'}\right)$ $xy' = 2y + \frac{x}{y'}$

Step 6: Further simplification to get the final differential equation.

Multiply both sides by $y'$ again: $x(y')^2 = 2yy' + x$ Rearrange the terms to match the given options: $x(y')^2 = x + 2yy'$

Step 7: Check if the solution matches the options.

Our derived differential equation is $x(y')^2 = x + 2yy'$. This matches option (D). However, we are given that the correct answer is option (A), which is $x(y')^2 = x - 2yy'$. Let's re-examine the steps to find the error.

Going back to Step 6: $xy' = 2y + \frac{x}{y'}$ Multiply both sides by $y'$: $x(y')^2 = 2yy' + x$ This is where the error was made. It should be: $x(y')^2 = 2yy' + x$ Now, let's solve for the correct answer. We have: $x^2 = 4b(y+b)$ $x = 2by' \implies b = \frac{x}{2y'}$ Substitute into the original equation: $x^2 = 4\left(\frac{x}{2y'}\right)\left(y + \frac{x}{2y'}\right)$ $x^2 = \frac{2x}{y'}\left(y + \frac{x}{2y'}\right)$ Divide by $x$: $x = \frac{2}{y'}\left(y + \frac{x}{2y'}\right)$ $xy' = 2y + \frac{x}{y'}$ Multiply by $y'$: $x(y')^2 = 2yy' + x$ Rearrange to get $x(y')^2 = x - 2yy'$, as in option (A). There must be a sign error somewhere. Let us try another approach. $x^2 = 4b(y+b) = 4by + 4b^2$ Differentiate with respect to x: $2x = 4by'$ $b = \frac{x}{2y'}$ Substituting: $x^2 = 4\left(\frac{x}{2y'}\right)y + 4\left(\frac{x}{2y'}\right)^2$ $x^2 = \frac{2xy}{y'} + \frac{x^2}{(y')^2}$ Divide by x: $x = \frac{2y}{y'} + \frac{x}{(y')^2}$ Multiply by $(y')^2$: $x(y')^2 = 2yy' + x$

We are looking for $x(y')^2 = x - 2yy'$. Let's try to force this result. Given $x^2 = 4b(y+b)$ Differentiating: $2x = 4b y'$ $b = \frac{2x}{4y'} = \frac{x}{2y'}$ Substituting back into original equation: $x^2 = 4(\frac{x}{2y'})(y + \frac{x}{2y'})$ $x^2 = \frac{2x}{y'}(y + \frac{x}{2y'})$ $x = \frac{2y}{y'} + \frac{x}{(y')^2}$ $x(y')^2 = 2y y' + x$

THERE IS AN ERROR IN THE PROBLEM. The correct answer should be x(y')^2 = x + 2yy'.

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs during algebraic manipulations, especially when substituting and rearranging equations.
• Division by Zero: Be mindful of potential division by zero and consider any necessary restrictions on variables.
• Verification: Always verify your final differential equation by substituting it back into the original equation (if possible) or by checking if it satisfies the given conditions.

Summary

The objective was to find the differential equation of the family of curves $x^2 = 4b(y+b)$. We differentiated the equation with respect to $x$, solved for the arbitrary constant $b$, and substituted it back into the original equation. After simplification, we arrived at the differential equation $x(y')^2 = x + 2yy'$. However, the given correct answer is x(y')^2 = x - 2yy', meaning there is an error.

Final Answer

The correct answer should be $x(y')^2 = x + 2yy'$. However, given the provided correct answer, this would correspond to option (D).

Since the question states option (A) is correct, and our derivation arrives at option (D), there is an error in the question itself. We are forced to assume there is an error in the question. We can rewrite the equation to match the provided correct answer as follows:

$x(y')^2 = x + 2yy'$ $x(y')^2 = x - 2yy'$

This is not possible, and there is an error in the problem.

Final Answer: There is an error in the problem. The correct answer should be x(y')^2 = x + 2yy', which corresponds to option (D). Since the question states that option (A) is correct, we must conclude there is an error in the problem or the answer key.