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

Showing the 12 most recent of 24 on this concept.

1. CBSE 2026Set ANNUAL1 mark
   Q.Write the degree of the differential equation $(y''')^2+3(y'')^3+3xy'+5y=0$

   ›Reveal solutionSolution

   The degree is the power of the highest-order derivative, once the equation is a polynomial in derivatives (no radicals/fractions of derivatives).

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

   The highest-order derivative present is $y'''$ (third order), and it already appears as a polynomial term.

   The power to which $y'''$ (the highest order derivative) is raised is $2$.

   ✓Final answer

   Degree $=2$

2. CBSE 2026Set ANNUAL1 markMCQ
   Q.State whether the following statement is true or false: Order and degree of a differential equation are always positive integers.

   (a) True

   (b) False

   ›Reveal solutionSolution

   Order = highest derivative order and degree = power of the highest-order derivative (equation cleared of radicals/fractions in the derivatives); each is a positive integer, so the statement is True.

   By definition, the order of a differential equation is the order of the highest derivative it contains — this is always a whole counting number, i.e. a positive integer ($1, 2, 3, \dots$).

   The degree is the power (exponent) of the highest-order derivative after the equation has been expressed as a polynomial in its derivatives (free of radicals and fractional powers of the derivatives). When it exists in this standard polynomial form, the degree is likewise a positive integer.

   For example, $\left(\dfrac{d^2y}{dx^2}\right)^3 + \dfrac{dy}{dx} = 0$ has order $2$ and degree $3$ — both positive integers.

   ✓Final answer

   The statement is True — option (i).

3. CBSE 2025Set ANNUAL1 mark
   Q.Write the order of the differential equation $\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=\left(\dfrac{d^2y}{dx^2}\right)^{3/2}$

   ›Reveal solutionSolution

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

   The given differential equation is

   $\sqrt{1+\left(\frac{dy}{dx}\right)^2}=\left(\frac{d^2y}{dx^2}\right)^{3/2}$

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

   ✓Final answer

   Order $=2$.

4. CBSE 2025Set ANNUAL1 markMCQ
   Q.The order and degree of the differential equation $\sqrt{\dfrac{dy}{dx}}-4\dfrac{dy}{dx}-7x=0$ are respectively :

   (a) $1, 2$

   (b) $2, 1$

   (c) $2, 2$

   (d) $1, 1$

   ›Reveal solutionSolution

   Isolating the radical and squaring turns the equation into a polynomial in $dy/dx$; the power of the highest derivative after clearing the radical gives the degree.

   1. Given $\sqrt{\dfrac{dy}{dx}}-4\dfrac{dy}{dx}-7x=0$. Let $p=\dfrac{dy}{dx}$.
   2. Isolate the radical: $\sqrt p=4p+7x$.
   3. Square both sides to remove the radical (a fractional power of a derivative is not allowed for degree to be defined): $p=(4p+7x)^2=16p^2+56px+49x^2$.
   4. Rearranged: $16p^2+(56x-1)p+49x^2=0$ — a polynomial equation in $p=\dfrac{dy}{dx}$.
   5. Order = highest order of derivative present = $1$ (only $\dfrac{dy}{dx}$ appears, no higher derivative).
   6. Degree = power of the highest-order derivative once the equation is a polynomial free of radicals/fractional powers of derivatives = $2$ (from the $p^2$ term).

   ✓Final answer

   (a) $1, 2$

5. CBSE 2025Set ANNUAL1 markMCQ
   Q.The order and degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = a^x$ are ______ respectively.

   (a) 1, 1

   (b) 1, 2

   (c) 2, 2

   (d) 2, 1

   ›Reveal solutionSolution

   The highest derivative is $\frac{d^2y}{dx^2}$ (order 2), and it appears raised to power 2 (degree 2).

   The equation is

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

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

   The equation is already polynomial in the derivatives (the right side $a^x$ contains no derivatives). The power to which the highest-order derivative $\dfrac{d^2y}{dx^2}$ is raised is $2$, so the degree is 2.

   ✓Final answer

   The order and degree are $2$ and $2$ respectively — the third option.

6. CBSE 2025Set MARCH1 markMCQ
   Q.The differential equation $\left(\dfrac{dx}{dy}\right)^{3}+2y^{1/2}=x$ is :

   (a) of order 1 and degree 6

   (b) of order 2 and degree 1

   (c) of order 1 and degree 2

   (d) of order 1 and degree 3

   ›Reveal solutionSolution

   The only derivative present is $\dfrac{dx}{dy}$ (order $1$), raised to power $3$ in a polynomial form, so the degree is $3$; option (d).

   Order. The equation $\left(\dfrac{dx}{dy}\right)^{3}+2y^{1/2}=x$ contains only the first derivative $\dfrac{dx}{dy}$, so the order is $1$.

   Degree. Degree is the highest power of the highest-order derivative when the equation is polynomial in its derivatives. Here that derivative appears as $\left(\dfrac{dx}{dy}\right)^{3}$.

   The radical $y^{1/2}$ involves the dependent variable, not a derivative, so it does not change the degree. Hence the degree is $3$.

   ✓Final answer

   Option (d) order $1$ and degree $3$.

7. CBSE 2024Set ANNUAL1 markMCQ
   Q.Order and degree of the differential equation y = dy/dx + c/(dy/dx) are

   (a) 1, 2

   (b) 2, 2

   (c) 1, 1

   (d) 2, 1

   ›Reveal solutionSolution

   Rewrite the equation as a polynomial in $dy/dx$; the highest power of the highest-order derivative gives the degree.

   Let $p = dy/dx$. The equation $y = p + c/p$ becomes, after multiplying through by $p$:

   $yp = p^2 + c \quad\Rightarrow\quad p^2 - yp + c = 0$

   This is a polynomial equation in $p = dy/dx$. The highest-order derivative present is the first derivative, so the order is 1. The highest power to which that derivative is raised (after clearing the fraction) is $2$, so the degree is 2.

   ✓Final answer

   Order $1$, degree $2$ — option (a).

8. CBSE 2024Set ANNUAL1 markMCQ
   Q.State whether the following statement is true or false: Order and degree of a differential equation are always positive integers.

   (a) True

   (b) False

   ›Reveal solutionSolution

   The order of a differential equation is always a positive integer, and its degree — whenever it exists — is also a positive integer, so the statement is taken as True.

   • Order = the order of the highest-order derivative appearing in the equation. Since we count derivatives ($\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, …), the order is always a positive integer such as $1, 2, 3, \dots$
   • Degree = the power of the highest-order derivative once the equation is written as a polynomial in the derivatives (free of radicals and fractions in the derivatives). When it is defined, this power is a positive integer.

   Both quantities are therefore positive integers, matching the standard textbook statement.

   ✓Final answer

   True. (Note: the degree must first be defined — i.e. the equation is a polynomial in its derivatives; where it exists it is a positive integer.)

9. CBSE 2024Set MARCH1 markMCQ
   Q.The order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}}=\sqrt{\frac{dy}{dx}+5}$ are respectively :

   (a) 2 and 1

   (b) 2 and 3

   (c) 2 and 2

   (d) 3 and 2

   ›Reveal solutionSolution

   Square both sides $\Rightarrow \frac{d^2y}{dx^2}=\frac{dy}{dx}+5$; order $2$, degree $1$.

   Starting from $\sqrt{\dfrac{d^2y}{dx^2}}=\sqrt{\dfrac{dy}{dx}+5}$, square both sides to clear the radicals:

   $\frac{d^2y}{dx^2}=\frac{dy}{dx}+5.$

   The highest-order derivative present is $\dfrac{d^2y}{dx^2}$ (order $2$), and after the equation is made rational and integral its power is $1$ (degree $1$).

   ✓Final answer

   Option (a) 2 and 1.

10. CBSE 2023Set ANNUAL1 mark
    Q.Write the degree of the differential equation $e^{\frac{dy}{dx}} + \dfrac{dy}{dx} = x$

    ›Reveal solutionSolution

    The derivative appears inside an exponential (transcendental) term, so the equation cannot be written as a polynomial in $\frac{dy}{dx}$.

    Degree is defined only when the differential equation can be expressed as a polynomial in the derivatives. Here $\frac{dy}{dx}$ occurs as the exponent of $e$ (a non-polynomial/transcendental form), so this cannot be reduced to polynomial form. Hence the degree of this differential equation is not defined.

    ✓Final answer

    Not defined

11. CBSE 2023Set ANNUAL1 markMCQ
    Q.Order of the differential equation (d²y/dx²)² = 1 + (dy/dx)³ is

    (a) 1

    (b) 3

    (c) 2

    (d) 4

    ›Reveal solutionSolution

    The order of a differential equation is the order of the highest derivative that appears in it — degree (the power it's raised to) is a separate idea.

    Step 1. The equation is $\left(\dfrac{d^2y}{dx^2}\right)^2 = 1+\left(\dfrac{dy}{dx}\right)^3$.

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

    Step 3. Hence the order is $2$ (its degree, incidentally, is $2$ since it appears squared after clearing radicals/fractions — but the question asks only for order).

    ✓Final answer

    Order $=2$ (option c).

12. CBSE 2023Set ANNUAL1 markMCQ
    Q.The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 = a^x$ is 3.

    (a) True

    (b) False

    ›Reveal solutionSolution

    The highest-order derivative is $\dfrac{d^2y}{dx^2}$, raised to power $2$, so the degree is $2$ — the claim of $3$ is False.

    The degree of a differential equation is the power to which the highest-order derivative is raised, once the equation is expressed as a polynomial in its derivatives (free of radicals and fractional powers).

    The equation is:

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

    • The highest-order derivative is $\dfrac{d^2y}{dx^2}$ (order $2$).
    • Its power in the equation is $2$.

    So the degree is $2$. The exponent $3$ belongs to the lower-order term $\dfrac{dy}{dx}$ and does not decide the degree. Hence the statement that the degree is $3$ is wrong.

    ✓Final answer

    False — the order is $2$ and the degree is $2$.