Q.The order and degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^3-3\frac{d^3y}{dx^3}+2\left(\frac{dy}{dx}\right)^4=y^4$ are:
(A) 1, 4
(B) 3, 4
(C) 2, 4
(D) 3, 2

The order is the highest derivative present; here it's $\dfrac{d^2y}{dx^2}$, so order $=2$.

The order of a differential equation is defined as the order of the highest-order derivative appearing in it.

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Order is the highest derivative present ($2$); degree is not defined because, once radicals are removed, the equation still contains $\cos(dy/dx)$, so it cannot be written as a polynomial in the derivatives.

The given differential equation is

$\frac{d^2y}{dx^2} = \sqrt{\cos\frac{dy}{dx}}.$

Order: The highest order derivative appearing is $\dfrac{d^2y}{dx^2}$ (a second-order derivative), so the order is 2.

Degree: The degree of a differential equation is defined only when the equation can be expressed as a polynomial in the derivatives $y', y'', \dots$ (after removing radicals and fractional powers involving the derivatives). Squaring both sides here gives …

The order of a differential equation is the order of the highest derivative present, regardless of any power/degree on that derivative — here the highest derivative is $\dfrac{d^4y}{dx^4}$, so the order is 4.

The given equation is

$\left(\frac{d^4y}{dx^4}\right)^5 + \sin(y'') = 0.$

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The family $y=a\sin(x+b)$ has 2 arbitrary constants, so eliminating them needs a 2nd-order ODE.

The order of the differential equation representing a family of curves equals the number of independent arbitrary constants in the family (since each differentiation can eliminate at most one constant).

Here $y=a\sin(x+b)$ contains two arbitrary constants, $a$ and $b$. To eliminate both, we differentiate twice: …

The highest derivative present is $\dfrac{d^2y}{dx^2}$ (order 2), and it occurs to the power $1$ (degree 1) in this already-polynomial equation.

The differential equation is

$\frac{d^2y}{dx^2} - 7\left(\frac{dy}{dx}\right)^3 + 6y = 0.$

Order = the order of the highest derivative appearing $= 2$ (from $\dfrac{d^2y}{dx^2}$).

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Highest derivative is $\dfrac{d^3y}{dx^3}$ (order 3); it appears to power 1, so degree 1.

The order of a differential equation is the order of the highest derivative present. Here the highest derivative is $\dfrac{d^3y}{dx^3}$, so the order is $3$.

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