Using the product rule of differentiation: $\frac{d}{dx}[u \cdot v] = u' v + u v'$
Here $u = x^2 \implies u' = 2x$
$v = \cos x \implies v' = -\sin x$

Order and Degree of a Differential Equation:

• Definition of Order: The order of a differential equation is the order of the highest derivative appearing in the equation.
• Given Equation:
  $$\left(\frac{d^2 y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + y^4 = 0$$
• The derivatives present are $\frac{dy}{dx}$ (first derivative, order 1) and $\frac{d^2 y}{dx^2}$ (second derivative, order 2).
• The highest order derivative is $\frac{d^2 y}{dx^2}$, which has an order of $2$ (its degree is also $2$).

Relations Between Roots and Coefficients of a Quadratic Equation:
For any quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$:

• Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
• Product of the roots: $\alpha \beta = \frac{c}{a}$