Find the differential equation of the followings where 𝑎, 𝑏 and 𝑐 are arbitrary
constant.    𝑦^2 = 4𝑎(𝑥 − 𝑏)

Given the equation:

\[
y^2 = 4a(x - b)
\]

where \(a\), \(b\), and \(c\) are arbitrary constants, we are tasked with finding the differential equation. Let's proceed step by step.

### Step 1: Implicit Differentiation
To find the differential equation, we first differentiate both sides of the equation with respect to \(x\).

The given equation is:

\[
y^2 = 4a(x - b)
\]

Taking the derivative of both sides with respect to \(x\), we apply the chain rule on the left-hand side:

\[
2y \frac{dy}{dx} = 4a
\]

This is a first-order differential equation. We can express \(\frac{dy}{dx}\) as:

\[
\frac{dy}{dx} = \frac{2a}{y}
\]

### Step 2: Eliminate the constants
Now, to eliminate the constant \(a\), we can go back to the original equation:

\[
y^2 = 4a(x - b)
\]

Solving for \(a\), we get:

\[
a = \frac{y^2}{4(x - b)}
\]

Substitute this expression for \(a\) into the differential equation \(\frac{dy}{dx} = \frac{2a}{y}\):

\[
\frac{dy}{dx} = \frac{2}{y} \cdot \frac{y^2}{4(x - b)}
\]

Simplify:

\[
\frac{dy}{dx} = \frac{y}{2(x - b)}
\]

### Step 3: Simplify the result
Finally, multiplying both sides by \(2(x - b)\), we get the differential equation:

\[
2(x - b) \frac{dy}{dx} = y
\]

This is the required differential equation that corresponds to the given relation \(y^2 = 4a(x - b)\).

---

Would you like further clarifications or details on any step?

Here are some related questions to explore:
1. What is the general solution to the differential equation \(2(x - b) \frac{dy}{dx} = y\)?
2. How can we use substitution to simplify solving first-order differential equations?
3. What are some typical methods for solving differential equations involving implicit functions?
4. How do boundary conditions affect the solution of a differential equation like this one?
5. How can the method of separation of variables be applied to this kind of equation?

**Tip**: When finding differential equations from implicit functions, carefully applying the chain rule and eliminating constants can simplify the process significantly.

Find the differential equation of the followings where 𝑎, 𝑏 and 𝑐 are arbitrary constant. 𝑦^2 = 4𝑎(𝑥 − 𝑏)

Explore the step-by-step solution to finding the differential equation for the given implicit function y^2 = 4a(x - b). Learn how to apply implicit differentiation, eliminate constants, and derive the final first-order differential equation. This problem is suitable for students in Grades 11-12.

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