Where Mindful Learning Begins
Where Mindful Learning Begins
This Class 12 Maths NCERT Solutions Chapter 9 Ex 9.1 page covers all 12 questions, solved step-by-step — identifying the highest order derivative to find the order of a differential equation, and checking whether the equation is a polynomial in its derivatives to find the degree.
Questions 1–10 ask you to determine the order and degree of a range of differential equations, from straightforward polynomial forms to trickier ones where a derivative sits inside a trigonometric function — the exact situation where degree becomes not defined, a distinction CBSE tests almost every year. Questions 11 and 12 are multiple-choice versions of the same skill, reinforcing that order and degree, when defined, are always positive integers.
The highest order derivative present is \dfrac{d^4y}{dx^4}, so the order is 4.
For the degree to be defined, the equation must be a polynomial equation in all the derivatives that appear. Here y''' sits inside a sine function, \sin(y'''), so the equation is not a polynomial equation in its derivatives.
The highest order derivative present is y', so the order is 1.
The equation is a polynomial equation in y', and the highest power raised to y' is 1.
The highest order derivative present is \dfrac{d^2s}{dt^2}, so the order is 2.
The equation is a polynomial equation in \dfrac{d^2s}{dt^2} and \dfrac{ds}{dt}, and the highest power raised to \dfrac{d^2s}{dt^2} is 1.
The highest order derivative present is \dfrac{d^2y}{dx^2}, so the order is 2.
Here \dfrac{dy}{dx} sits inside a cosine function, \cos\left(\dfrac{dy}{dx}\right), so the equation is not a polynomial equation in its derivatives — the degree cannot be defined even though the order can.
The highest order derivative present is \dfrac{d^2y}{dx^2}, so the order is 2.
The equation is a polynomial equation in \dfrac{d^2y}{dx^2} (the trigonometric terms involve only x, not a derivative), and the highest power raised to \dfrac{d^2y}{dx^2} is 1.
The highest order derivative present is y''', so the order is 3.
The equation is a polynomial equation in y''', y'' and y', and the highest power raised to y''' is 2.
The highest order derivative present is y''', so the order is 3.
The equation is a polynomial equation in y''', y'' and y', and the highest power raised to y''' is 1.
The highest order derivative present is y', so the order is 1.
The equation is a polynomial equation in y' (the exponential term involves only x, not a derivative), and the highest power raised to y' is 1.
The highest order derivative present is y'', so the order is 2.
The equation is a polynomial equation in y'' and y', and the highest power raised to y'' is 1.
The highest order derivative present is y'', so the order is 2.
Here \sin y is a function of y itself, not of any derivative, so it doesn't affect whether the equation is a polynomial in its derivatives.
The equation is a polynomial equation in y'' and y', and the highest power raised to y'' is 1.
The highest order derivative present is \dfrac{d^2y}{dx^2}, giving order 2. But \dfrac{dy}{dx} appears inside a sine function, \sin\left(\dfrac{dy}{dx}\right).
Since the equation must be a polynomial equation in all the derivatives present — not just the highest order one — and it isn't a polynomial in \dfrac{dy}{dx}, the degree cannot be defined.
The highest order derivative present in the equation is \dfrac{d^2y}{dx^2}, a second order derivative.
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