Animated Solution for Mathematics - Differential Equations: The difference between degree and order of differential equation that represents the family of curves given by $y^2=a(x+\frac{\sqrt{2}}{2})$, $a>0$ is

We are examining the family of curves defined by the equation:

where $a>0$. This expression represents a collection of parabolas that vary based on the parameter $a$. Our objective is to determine the differential equation for this family and calculate the difference between its degree and its order.

In the study of differential equations, the order is determined by the number of independent arbitrary constants present in the original equation.

Our equation $y^2=a(x+\frac{\sqrt{a}}{2})$ contains only one arbitrary constant, which is $a$. Consequently, the order of the resulting differential equation must be $1$.

Eliminating the Constant

To derive the differential equation, we must eliminate $a$. We begin by differentiating the original equation with respect to $x$:

Letting $y^{\prime }=\frac{dy}{dx}$, we obtain the relation $a=2y{y}^{\prime }$. We now substitute this expression for $a$ back into the original equation:

Assuming $y eq 0$, we divide both sides by $y$ to simplify the equation:

Expanding the right side yields:

To prepare for the final form, we isolate the radical term:

To define the degree, the equation must be a polynomial in its derivatives. We eliminate the radical by squaring both sides:

Expanding the left side and simplifying the right side, we get:

The highest power of the derivative $y^{\prime }$ in this polynomial equation is $3$. Therefore, the degree of the differential equation is $3$.

The difference between the degree and the order is: