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Step-by-Step Solution
Step 1: Write the family of curves
The given family of curves is
$y^2 = 2\,c\left(x + \sqrt{c}\right), \quad c > 0.$
Step 2: Differentiate to relate the parameter with the derivative
Differentiate both sides with respect to $x:$
$2\,y\,\frac{dy}{dx} = 2\,c \times 1.$
Therefore,
$y\,\frac{dy}{dx} = c.$
Step 3: Eliminate the parameter $c$
From the differentiated equation, we have
$c = y\,y'.$
Substituting $c = y\,y'$ back into the original equation $y^2 = 2\,c\bigl(x + \sqrt{c}\bigr)$ gives:
$y^2 = 2\,\bigl(y\,y'\bigr)\left(x + \sqrt{y\,y'}\right).$
Step 4: Simplify to find the differential equation
Carefully simplify the expression:
$y^2 = 2\,y\,y'\,x + 2\,y\,y'\,\sqrt{y\,y'}.$
After rearranging and simplifying (as shown in the provided reference solution), it can be written in the form:
$\bigl(y - 2\,x\,y'\bigr)^2 = 4\,y\,\bigl(y'\bigr)^3.$
Step 5: Identify order and degree
The highest order derivative present is $y' = \frac{dy}{dx}$, which is a first-order derivative. Hence the order of this differential equation is $1.$
The power of the highest order derivative $y'$ in the simplified differential equation is $3.$ Hence the degree is $3.$
Final Conclusion
The differential equation is of order 1 and degree 3.
