Q.The sum of the order and the degree of the differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = \sin y$ is :
(A) 5
(B) 2
(C) 3
(D) 4

Showing the 12 most recent of 93 on this concept.

1. CBSE 2026Set 65/1/11 markMCQ
   Q.The order and degree of the differential equation d dx(ey) = 0 respectively are 1 (A) 0, 1 (B) 1, 1 (C) 2, 1 (D) 1, not defined

   ›Reveal solutionSolution

   The given equation $\frac{d}{dx}(e^y)=0$ simplifies to $e^y \frac{dy}{dx}=0$, which is a first-order differential equation. Since $e^y \neq 0$, the highest derivative is $\frac{dy}{dx}$ raised to the power 1, so the degree is 1. The correct option is (B).

   The order of a differential equation is the highest order derivative present. The degree is the power of the highest order derivative, provided the equation is polynomial in derivatives. Here, the equation looks deceptively simple — but we must first expand it properly.

   1. Expand the derivative. The given equation is $\frac{d}{dx}(e^y) = 0$. Using the chain rule:

   $\frac{d}{dx}(e^y) = e^y \cdot \frac{dy}{dx}.$

   So the equation becomes:

   $e^y \frac{dy}{dx} = 0.$

   2. Identify the highest derivative.

      The only derivative present is $\frac{dy}{dx}$, which is a first derivative. Hence the order is $1$.

   3. Determine the degree.

      The degree is defined only when the differential equation is a polynomial in the derivatives. Here, the term $e^y$ is not a polynomial in $y$ or its derivatives — it's an exponential function of $y$. However, the derivative $\frac{dy}{dx}$ itself appears with power 1, and the equation is already in the form $e^y \cdot \frac{dy}{dx} = 0$.

      Since $e^y$ is never zero for any real $y$, we can divide both sides by $e^y$ to get:

   $\frac{dy}{dx} = 0.$

   This is a polynomial in $\frac{dy}{dx}$ (specifically, it is $\left(\frac{dy}{dx}\right)^1 = 0$). So the degree is $1$.

   Watch out

   A common mistake is to think that because $e^y$ is not a polynomial, the degree is "not defined". But the degree is defined for the highest derivative term — here $\frac{dy}{dx}$ appears with power 1, and the equation can be simplified to a polynomial form. So degree is defined and equals 1.

   Tip

   If the equation had been something like $\frac{d}{dx}(\sin y) = 0$, the same logic applies: $\cos y \cdot \frac{dy}{dx} = 0$ simplifies to $\frac{dy}{dx} = 0$ (since $\cos y$ is not identically zero), giving order 1, degree 1.

   4. Match with the options. Order = 1, Degree = 1 corresponds to option (B).

   ✓Final answer

   The order is 1 and the degree is 1, so the correct option is (B).

2. CBSE 2026Set V11 mark
   Q.Choose from $[0, 3, -1, 2, -2, 1]$. If $m$ and $n$ are respectively the order and degree of the differential equation $2x^2\frac{d^2y}{dx^2}-3\frac{dy}{dx}+y=0$ then $m+n=$ ____.

   ›Reveal solutionSolution

   Order $m=2$ and degree $n=1$ give $m+n=3$.

   In $2x^2\dfrac{d^2y}{dx^2}-3\dfrac{dy}{dx}+y=0$:

   • The highest-order derivative is $\dfrac{d^2y}{dx^2}$, so the order is $m=2$.
   • The equation is polynomial in the derivatives and the highest derivative occurs to the first power, so the degree is $n=1$.

   Therefore $m+n=2+1=3$.

   ✓Final answer

   $3$

3. CBSE 2026Set A1 markMCQ
   Q.The order and degree of the differential equation $\left(\frac{d^2 s}{dt^2}\right)^2 + \left(\frac{ds}{dt}\right)^3 + 4 = 0$ is

   (a) order = 2, degree = 1

   (b) order = 2, degree = 2

   (c) order = 1, degree = 2

   (d) order = 1, degree = 1

   ›Reveal solutionSolution

   Highest derivative is second order; it appears squared → order 2, degree 2.

   The order is the order of the highest derivative present. Here $\dfrac{d^2 s}{dt^2}$ is the highest, so order $=2$.

   The degree is the power of that highest-order derivative after the equation is made polynomial in derivatives. Since $\left(\dfrac{d^2 s}{dt^2}\right)^2$ appears, the degree $=2$.

   ✓Final answer

   (B) order = 2, degree = 2.

4. CBSE 2026Set ANNUAL1 markMCQ
   Q.What is the degree of the differential equation $\frac{d^4y}{dx^4}-\sin\left(\frac{d^3y}{dx^3}\right)=0$?

   (a) 4

   (b) 3

   (c) 0

   (d) Undefined

   ›Reveal solutionSolution

   The degree of a differential equation is defined only when it can be written as a polynomial in derivatives; here a $\sin(\cdot)$ of a derivative appears, so the degree is undefined.

   The degree of a differential equation is the highest power of the highest-order derivative, provided the equation is a polynomial in all its derivatives (after clearing any fractional powers/roots and radicals).

   Here the equation is

   $\frac{d^4y}{dx^4}-\sin\left(\frac{d^3y}{dx^3}\right)=0$

   The term $\sin\left(\dfrac{d^3y}{dx^3}\right)$ cannot be expressed as a polynomial (finite power) in $\dfrac{d^3y}{dx^3}$ — it is a transcendental function of a derivative. Since the equation is not a polynomial equation in the derivatives, its degree is not defined.

   (Its order, however, is well defined: the highest derivative present is $\frac{d^4y}{dx^4}$, so the order is 4.)

   ✓Final answer

   The correct option is (d) Undefined (the degree is not defined).

5. CBSE 2026Set ANNUAL1 markMCQ
   Q.The order of the differential equation $2x^2 \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0$ is

   (a) $2$

   (b) $1$

   (c) $0$

   (d) Not defined

   ›Reveal solutionSolution

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

   The equation $2x^2\dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + y = 0$ contains $\dfrac{d^2y}{dx^2}$ as its highest derivative.

   ✓Final answer

   The correct option is (a) $2$.

6. CBSE 2026Set ANNUAL1 markMCQ
   Q.The order of the differential equation $\dfrac{d^2y}{dx^2} = \sqrt{1+\left(\dfrac{dy}{dx}\right)^2}$ is

   (a) 1

   (b) 2

   (c) 3

   (d) None of these

   ›Reveal solutionSolution

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

   The equation $\dfrac{d^2y}{dx^2} = \sqrt{1+\left(\dfrac{dy}{dx}\right)^2}$ contains $\dfrac{d^2y}{dx^2}$ (order 2) and $\dfrac{dy}{dx}$ (order 1). The highest is order 2.

   ✓Final answer

   (b) $2$.

7. CBSE 2026Set ANNUAL1 markMCQ
   Q.The degree of the differential equation $\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \sin\left(\dfrac{dy}{dx}\right) + 1 = 0$ is:

   (a) 3

   (b) 2

   (c) 1

   (d) Not defined

   ›Reveal solutionSolution

   Degree is defined only when the differential equation is a polynomial in derivatives; the $\sin(dy/dx)$ term breaks this.

   The degree of a differential equation is the power of the highest-order derivative, but this is only defined when the equation can be written as a polynomial in the derivatives (no transcendental functions of a derivative).

   Here $\left(\dfrac{d^2y}{dx^2}\right)^3+\left(\dfrac{dy}{dx}\right)^2+\sin\left(\dfrac{dy}{dx}\right)+1=0$ contains $\sin\left(\dfrac{dy}{dx}\right)$, which cannot be expressed as a polynomial in $\dfrac{dy}{dx}$.

   Hence the degree is not defined (the order, however, is 2).

   ✓Final answer

   Option (d): Not defined.

8. CBSE 2026Set ANNUAL1 markMCQ
   Q.Order of differential equation d²y/dx² - 2 dy/dx + 3y = 0 is:

   (a) 3

   (b) 2

   (c) 1

   (d) 0

   ›Reveal solutionSolution

   The order of a differential equation is the order of the highest derivative present in it.

   The given differential equation is:

   $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 3y = 0$

   The highest order derivative appearing is $\dfrac{d^2y}{dx^2}$, a second-order derivative.

   ✓Final answer

   The order of the differential equation is 2 (option b).

9. CBSE 2026Set ANNUAL1 markMCQ
   Q.Order and degree of differential equation \left(\dfrac{d^2y}{dx^2}\right)^3 = \left[y + \left(\dfrac{dy}{dx}\right)^2\right] are:

   (a) 2, 4

   (b) 2, 3

   (c) 2, 2

   (d) 1, 2

   ›Reveal solutionSolution

   Order = highest-order derivative present; degree = power of that highest-order derivative, once the equation is a polynomial in derivatives.

   Working: The equation is

   $\left(\frac{d^2y}{dx^2}\right)^3 = y + \left(\frac{dy}{dx}\right)^2$

   The highest-order derivative present is $\dfrac{d^2y}{dx^2}$ (a second derivative) $\Rightarrow$ order $=2$.

   Since the equation is already a polynomial in derivatives, and $\dfrac{d^2y}{dx^2}$ appears raised to the power $3$, the degree $=3$.

   ✓Final answer

   Order $= 2$, degree $= 3$ — option (b).

10. CBSE 2026Set ANNUAL1 markMCQ
    Q.Order and degree of differential equation (y''')² + (y'')³ + (y')⁴ + y⁵ = 0 is

    (a) Order = 2, Degree = 3

    (b) Order = 3, Degree = 2

    (c) Order = 1, Degree = 4

    (d) Order = 3, Degree = 3

    ›Reveal solutionSolution

    Order is the index of the highest derivative present; degree is the power of that highest derivative once the equation is polynomial in derivatives.

    Given: $(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0$.

    Order: the highest-order derivative appearing is $y'''$ (the third derivative), so the order is 3.

    Degree: the equation is already a polynomial in the derivatives (no radicals/fractional powers of derivatives to clear), and the power to which the highest-order derivative $y'''$ is raised is $2$. So the degree is 2.

    ✓Final answer

    Order = 3, Degree = 2. (Option b)

11. CBSE 2026Set ANNUAL1 markMCQ
    Q.The order and degree of the differential equation $\left(\dfrac{dy}{dx}\right)^4 + 3y\dfrac{d^2y}{dx^2} = 0$ are respectively

    (a) 2 and 4

    (b) 4 and 2

    (c) 2 and 1

    (d) 1 and 2

    ›Reveal solutionSolution

    The order is the highest derivative present; the degree is the power of that highest-order derivative once the equation is written as a polynomial in the derivatives.

    Given: $\left(\dfrac{dy}{dx}\right)^4 + 3y\dfrac{d^2y}{dx^2} = 0$

    Order: The highest-order derivative appearing is $\dfrac{d^2y}{dx^2}$, a second-order derivative. So order $=2$.

    Degree: The equation is already a polynomial in the derivatives $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ (no fractional/negative powers, no derivative inside a transcendental function). The highest-order derivative, $\dfrac{d^2y}{dx^2}$, occurs to the power $1$. So degree $=1$.

    (Note: the power $4$ on $\dfrac{dy}{dx}$ does not affect the degree — degree only looks at the power of the highest-order derivative term.)

    ✓Final answer

    (c) 2 and 1

12. CBSE 2026Set ANNUAL1 markMCQ
    Q.The number of arbitrary constants in the general solution of a differential equation of third order will be:

    (a)

    (i) 0

    (b)

    (ii) 1

    (c)

    (iii) 2

    (d)

    (iv) 3

    ›Reveal solutionSolution

    A third-order differential equation has $3$ arbitrary constants in its general solution — option (iv).

    Concept. The order of a differential equation is the order of its highest derivative, and the general solution contains exactly as many independent arbitrary constants as the order (each integration introduces one constant).

    Steps.

    • Order of the equation $=3$.

    • Solving requires (in effect) integrating three times, introducing three independent arbitrary constants.

    • Hence the general solution has $3$ arbitrary constants.

    ✓Final answer

    $3$ arbitrary constants — option (iv).