For each of the differential equations given below, indicates its order and degree (if defined).
\((i) \frac {d^2y}{dx^2}+5x(\frac {dy}{dx})^2-6y=log\ x\)
\((ii)(\frac {dy}{dx})^3-4(\frac{dy}{dx})^2+7y=sin\ x\)
\((iii) \frac {d^4y}{dx^4}-sin(\frac {d^3y}{dx^3})=0\)
(i) The differential equation is given as:
\(\frac {d^2y}{dx^2}+5x(\frac {dy}{dx})^2-6y=log\ x\)
⇒\(\frac {d^2y}{dx^2}+5x(\frac {dy}{dx})^2-6y-log\ x=0\)
The highest order derivative present in the differential equation is \(\frac {d^2y}{dx^2}\).Thus, its order is two.The highest power raised to \(\frac {d^2y}{dx^2}\) is one. Hence, its degree is one.
(ii) The differential equation is given as:
\((\frac {dy}{dx})^3-4(\frac{dy}{dx})^2+7y=sin\ x\)
⇒\((ii)(\frac {dy}{dx})^3-4(\frac{dy}{dx})^2+7y-sin\ x=0\)
The highest order derivative present in the differential equation is dy/dx. Thus, its order is one.The highest power raised to \(\frac {dy}{dx}\) is three.Hence, its degree is three.
(iii) The differential equation is given as:
\(\frac {d^4y}{dx^4}-sin(\frac {d^3y}{dx^3})=0\)
The highest order derivative present in the differential equation is \(\frac {d^4y}{dx^4}\). Thus, its order is four. However, the given differential equation is not a polynomial equation. Hence, its degree is not defined.
Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f(x + y) = f(x) f(y) \) for all \( x, y \in \mathbb{R} \). If \( f'(0) = 4a \) and \( f \) satisfies \( f''(x) - 3a f'(x) - f(x) = 0 \), where \( a > 0 \), then the area of the region R = {(x, y) | 0 \(\leq\) y \(\leq\) f(ax), 0 \(\leq\) x \(\leq\) 2\ is :