Understanding the Differential Equation from a Family of Curves

The problem asks for the degree of the differential equation derived from the given family of curves:

\begin{equation*} y^2 = 2cx + 2c\sqrt{c} \end{equation*}

where $c$ is a positive parameter. To find the differential equation, we must eliminate the parameter $c$. We can do this by differentiating the equation with respect to $x$ and then substituting the expression for $c$ back into the original equation.

Eliminating the Parameter 'c'

Differentiate the given equation with respect to $x$:

\begin{equation*} \frac{d}{dx}(y^2) = \frac{d}{dx}(2cx + 2c\sqrt{c}) \end{equation*}

Using the chain rule on the left side and treating $c$ as a constant with respect to $x$ (as it's a parameter of the curve family), we get:

\begin{equation*} 2y \frac{dy}{dx} = 2c \cdot \frac{d}{dx}(x) + \frac{d}{dx}(2c\sqrt{c}) \end{equation*}

Since $2c\sqrt{c}$ is a constant with respect to $x$, its derivative is zero. The derivative of $x$ with respect to $x$ is 1.

\begin{equation*} 2y \frac{dy}{dx} = 2c(1) + 0 \end{equation*}

\begin{equation*} 2y \frac{dy}{dx} = 2c \end{equation*}

Solving for $c$, we get:

\begin{equation*} c = y \frac{dy}{dx} \end{equation*}

Now, substitute this expression for $c$ back into the original equation $y^2 = 2cx + 2c\sqrt{c}$:

\begin{equation*} y^2 = 2 \left( y \frac{dy}{dx} \right) x + 2 \left( y \frac{dy}{dx} \right) \sqrt{y \frac{dy}{dx}} \end{equation*}

Let $p = \frac{dy}{dx}$ for simplicity. The equation becomes:

\begin{equation*} y^2 = 2(yp)x + 2(yp)\sqrt{yp} \end{equation*}

\begin{equation*} y^2 = 2xyp + 2yp\sqrt{yp} \end{equation*}

Simplifying the Differential Equation to Find the Degree

To determine the degree of the differential equation, we need to remove any radicals involving derivatives. In this equation, we have a square root term $\sqrt{yp}$.

Isolate the term with the radical:

\begin{equation*} y^2 - 2xyp = 2yp\sqrt{yp} \end{equation*}

Square both sides of the equation to eliminate the square root:

\begin{equation*} (y^2 - 2xyp)^2 = (2yp\sqrt{yp})^2 \end{equation*}

\begin{equation*} (y^2 - 2xyp)^2 = (2yp)^2 (\sqrt{yp})^2 \end{equation*}

\begin{equation*} (y^2 - 2xyp)^2 = 4y^2 p^2 (yp) \end{equation*}

\begin{equation*} (y^2 - 2xyp)^2 = 4y^3 p^3 \end{equation*}

Expanding the left side gives:

\begin{equation*} (y^2)^2 - 2(y^2)(2xyp) + (2xyp)^2 = 4y^3 p^3 \end{equation*}

\begin{equation*} y^4 - 4xy^3 p + 4x^2 y^2 p^2 = 4y^3 p^3 \end{equation*}

Replacing $p$ with $\frac{dy}{dx}$, the differential equation is:

\begin{equation*} y^4 - 4xy^3 \left(\frac{dy}{dx}\right) + 4x^2 y^2 \left(\frac{dy}{dx}\right)^2 = 4y^3 \left(\frac{dy}{dx}\right)^3 \end{equation*}

This can be written as:

\begin{equation*} 4y^3 \left(\frac{dy}{dx}\right)^3 - 4x^2 y^2 \left(\frac{dy}{dx}\right)^2 + 4xy^3 \left(\frac{dy}{dx}\right) - y^4 = 0 \end{equation*}

Determining the Order and Degree

The order of a differential equation is the order of the highest derivative present in the equation. In this equation, the highest derivative is $\frac{dy}{dx}$, which is a first-order derivative. Therefore, the order of the differential equation is 1.

The degree of a differential equation is the power of the highest order derivative, provided the equation has been made free from radicals and fractions as far as the derivatives are concerned. We have already cleared the radical in the previous step.

In the equation $4y^3 \left(\frac{dy}{dx}\right)^3 - 4x^2 y^2 \left(\frac{dy}{dx}\right)^2 + 4xy^3 \left(\frac{dy}{dx}\right) - y^4 = 0$, the highest order derivative is $\frac{dy}{dx}$. The powers of $\frac{dy}{dx}$ present are 1, 2, and 3. The highest power among these is 3.

Therefore, the degree of the differential equation is 3.

Conclusion

The differential equation representing the family of curves $y^2 = 2cx + 2c\sqrt{c}$ has a degree of 3.

Revision Table: Differential Equation Properties

Understanding the key properties like order and degree is crucial for classifying and solving differential equations.

• Order: The order of the highest derivative appearing in the equation.
• Degree: The power of the highest order derivative, after the equation is rationalized (freed from radicals and fractions involving derivatives).

Additional Information: Forming Differential Equations

A differential equation can be formed from a given family of curves by eliminating the arbitrary constants (or parameters). The number of arbitrary constants in the equation of the family of curves determines the order of the resulting differential equation. In this problem, there is one parameter '$c$', leading to a first-order differential equation.

The steps typically involve:

1. Write down the given equation of the family of curves.
2. Differentiate the equation with respect to the independent variable (usually $x$) as many times as there are arbitrary constants.
3. Eliminate the arbitrary constants from the original equation and the equations obtained by differentiation. The resulting equation is the required differential equation.