Understanding Differential Equations and Order

The question asks for the order of the differential equation that represents a given family of curves. The family of curves is described by the equation \(y^2 = 2cx + 2c\sqrt{c}\), where \(c\) is a positive parameter.

The order of a differential equation is defined as the order of the highest derivative appearing in the equation. To find the differential equation representing a given family of curves, we need to eliminate the arbitrary parameters present in the equation of the family.

Eliminating the Parameter to Form the Differential Equation

The given equation for the family of curves is:

\(y^2 = 2cx + 2c\sqrt{c}\)

We can rewrite \(2c\sqrt{c}\) as \(2c^{3/2}\).

\(y^2 = 2cx + 2c^{3/2}\)

This equation contains one arbitrary parameter, \(c\). To eliminate one parameter, we typically need to differentiate the equation once with respect to the independent variable \(x\).

Step-by-Step Differentiation

We differentiate the equation \(y^2 = 2cx + 2c^{3/2}\) with respect to \(x\). Remember that \(y\) is considered a function of \(x\), and \(c\) is a constant parameter.

Applying the chain rule to \(y^2\):

\(\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}\)

Differentiating the term \(2cx\) with respect to \(x\):

\(\frac{d}{dx}(2cx) = 2c \frac{dx}{dx} = 2c\)

Differentiating the term \(2c^{3/2}\) with respect to \(x\):

\(\frac{d}{dx}(2c^{3/2}) = 0\), since \(c\) is a constant with respect to \(x\).

Now, we differentiate the entire equation:

\(\frac{d}{dx}(y^2) = \frac{d}{dx}(2cx) + \frac{d}{dx}(2c^{3/2})\)

\(2y \frac{dy}{dx} = 2c + 0\)

Using the notation \(y' = \frac{dy}{dx}\), this becomes:

\(2y y' = 2c\)

Dividing both sides by 2, we get an expression for \(c\) in terms of \(y\) and \(y'\):

\(c = y y'\)

Substituting Back to Obtain the Differential Equation

Now we substitute this expression for \(c\) back into the original equation of the family of curves \(y^2 = 2cx + 2c^{3/2}\) to eliminate the parameter \(c\).

Substitute \(c = yy'\):

\(y^2 = 2(yy')x + 2(yy')^{3/2}\)

\(y^2 = 2xyy' + 2(yy')\sqrt{yy'}\)

This equation \(y^2 = 2xyy' + 2(yy')^{3/2}\) is the differential equation that represents the given family of curves.

Determining the Order of the Differential Equation

The order of a differential equation is the order of the highest derivative that appears in the equation. Let's look at the equation we obtained:

\(y^2 = 2xyy' + 2(yy')^{3/2}\)

The only derivative present in this equation is \(y'\), which represents the first derivative \(\frac{dy}{dx}\).

Since the highest order derivative is the first derivative, the order of this differential equation is 1.

Conclusion: Order of the Differential Equation

By eliminating the single arbitrary parameter \(c\) from the equation of the family of curves \(y^2 = 2cx + 2c\sqrt{c}\), we obtained a differential equation where the highest derivative is the first derivative. Therefore, the order of the differential equation is 1.

Revision Table: Key Concepts in Differential Equations

Additional Information: Forming Differential Equations

The standard method for finding the differential equation that represents a family of curves involves differentiation. The number of times you need to differentiate is generally equal to the number of essential arbitrary constants in the family's equation.

• If a family of curves has \(n\) essential arbitrary constants, the differential equation representing it will typically be of order \(n\).
• The steps usually involve writing down the equation of the family, differentiating it repeatedly, and then using algebraic manipulation (including substituting derivatives back into the original or differentiated equations) to eliminate all the arbitrary constants.
• The resulting equation, free of arbitrary constants and involving derivatives, is the differential equation for the family.