The differential equation for the family of circle x² + y² - 2ay = 0, where a is an arbitrary consta

Q: Wait, why do we differentiate only once? Can't we differentiate again to make it simpler?

Great question! The order of a differential equation is tied to the number of arbitrary constants. If you differentiate more than once, you will introduce higher-order derivatives that do not belong to the original family. We must stop exactly when the constant is eliminated.

Q: Why does the $y^2$ term become $2yy'$? I always forget that.

That is the chain rule in action! When you differentiate $y^2$ with respect to $x$, you treat $y$ as a function of $x$. So, $\frac{d}{dx}(y^2) = \frac{d}{dy}(y^2) \cdot \frac{dy}{dx} = 2y \cdot y'$. Keep that chain rule close to your heart!

Q: Is there a geometric way to solve this without all this algebra?

Algebra is the language of geometry here. While you could use the slope of the tangent, the algebraic elimination method is the most robust and reliable way to ensure you capture the entire family of curves without missing any details.

Question

The differential equation for the family of circle $x^2 + y^2 - 2ay = 0$, where $a$ is an arbitrary constant is

Options:

(A) $(x^2 + y^2)y' = 2xy$

(B) $2(x^2 + y^2)y' = xy$

(C) $(x^2 - y^2)y' = 2xy$

(D) $2(x^2 - y^2)y' = xy$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are exploring a family of circles passing through the origin $(0,0)$ with centers located on the $y$-axis. The family is defined by the equation:

$$x^2 + y^2 - 2ay = 0$$

Here, the parameter $a$ acts as the heartbeat of the family, determining both the radius and the vertical position of the center $(0, a)$. Our objective is to derive the differential equation that governs this entire family.

### The Strategy: The Art of Elimination

In the study of differential equations, the number of arbitrary constants serves as our guide. Since we have only one constant, $a$, we only need to differentiate the equation once.

This process allows us to eliminate the parameter $a$ and reveal the underlying differential relationship. Think of it as a detective game: with one piece of evidence, we only need one interrogation to uncover the truth.

### The Execution: The Algebraic Dance

We begin by differentiating the equation $x^2 + y^2 - 2ay = 0$ with respect to $x$. Applying the chain rule, we obtain:

$$2x + 2y y' - 2a y' = 0$$

Where $y'$ represents the derivative $\frac{dy}{dx}$. Simplifying this by dividing by $2$, we get:

$$x + y y' - a y' = 0$$

To isolate $a$, we rearrange the terms:

$$a y' = x + y y' \quad \Rightarrow \quad a = \frac{x + y y'}{y'}$$

### The Final Synthesis

Now, we substitute this expression for $a$ back into our original equation, $x^2 + y^2 - 2ay = 0$:

$$x^2 + y^2 - 2y \left( \frac{x + y y'}{y'} \right) = 0$$

To clear the denominator, we multiply the entire equation by $y'$:

$$(x^2 + y^2)y' - 2y(x + y y') = 0$$

Expanding the terms, we arrive at:

$$(x^2 + y^2)y' - 2xy - 2y^2 y' = 0$$

Grouping the terms containing $y'$, we find:

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

The final differential equation defining our family of circles is:

$$(x^2 - y^2)y' = 2xy$$

This is the elegant, concise expression of the geometric reality we set out to capture.

Animated Solution for Mathematics - Differential Equations: The differential equation for the family of circle $x^{2} + y^{2} - 2 a y = 0$ , where $a$ is an arbitrary constant is

Select Answer:

Visualized Solution

Family of Circles

Geometric Interpretation

The Logic Bridge

Differentiating w.r.t $x$

Applying the Derivative

Simplifying the Equation

Isolating the Constant $a$

Expressing $a$

The Substitution Step

Raw Substitution

Clearing the Denominator

Expanding the Terms

Grouping $y^{'}$ Terms

Final Simplification

Conclusion

The Sigma Insight: Formation of Differential Equations

Analyzing the Setup

The Strategy

The Execution

The Final Synthesis

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Animated Solution for Mathematics - Differential Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The differential equation for the family of circle x2+y2−2ay=0, where a is an arbitrary constant is