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

Showing the 12 most recent of 13 on this concept.

1. GUJCET 2023Set 091 markMCQ
   Q.The order and degree of the differential equation $\sqrt[4]{\left(\frac{d^{3}y}{dx^{3}}\right)^{5}} = \sqrt[3]{\left(\frac{d^{2}y}{dx^{2}}\right)^{4}}$ is : (A) $2$ and $16$ (B) $3$ and $15$ (C) $3$ and $16$ (D) $2$ and $12$

   ›Reveal solutionSolution

   Order is the highest derivative; degree is its power after removing all radicals.

   Concept. $\sqrt[4]{(y''')^5}=\sqrt[3]{(y'')^4}$ means $(y''')^{5/4}=(y'')^{4/3}$. Highest derivative is $y'''$, so order $=3$.

   Solution. Raise both sides to the 12th power to clear fractional exponents: …

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2. GUJCET 2024Set 131 markMCQ
   Q.The order and the degree of the differential equation $\sqrt{\frac{d^2 y}{dx^2}} = \sqrt[3]{\left(\frac{dy}{dx}\right)^4} + 2$ is respectively __________ and __________. (A) $1, 8$ (B) $3, 2$ (C) $2, 8$ (D) $2, 3$

   ›Reveal solutionSolution

   Order = 2 (the $\tfrac{d^2y}{dx^2}$ term); after removing the fractional powers the degree of that highest derivative is 3.

   Order. The highest derivative present is $\dfrac{d^2y}{dx^2}$, so order $=2$. …

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3. GUJCET 2025Set 031 markMCQ
   Q.The degree of the differential equation $\left(1 + \frac{dy}{dx}\right)^{\frac{1}{2}} = \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{3}}$ is _____ (A) 4 (B) 2 (C) 3 (D) 1

   ›Reveal solutionSolution

   Make the equation polynomial in derivatives, then read the power of the highest-order derivative.

   Raise both sides of $\left(1+\tfrac{dy}{dx}\right)^{1/2}=\left(\tfrac{d^2y}{dx^2}\right)^{1/3}$ to the 6th power: …

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4. GUJCET 2020Set 071 markMCQ
   Q.The order and degree of differential equation $\left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{3/2} = \frac{d^2y}{dx^2}$ are $p$ and $q$ respectively then $p + q =$ ________. (A) $6$ (B) $4$ (C) $2$ (D) $5$

   ›Reveal solutionSolution

   Highest derivative is $y''$ (order 2); after squaring to remove the $3/2$ power, its power is 2 (degree 2), so $p+q=4$.

   Concept: Order = highest derivative present; degree = power of that highest derivative once the equation is a polynomial in derivatives.

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

   Square both sides to clear the $3/2$ power: …

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5. GUJCET 2021Set 151 markMCQ
   Q.Order and degree of the differential equation $e^{\frac{d^2y}{dx^2}} = x$ are $\underline{\hspace{2cm}}$ respectively. (A) 2 and not defined (B) 1 and 2 (C) 2 and 1 (D) 1 and not defined

   ›Reveal solutionSolution

   [!TLDR]

   Order $= 2$, degree $=$ not defined.

   Concept

   • Order $=$ order of the highest derivative appearing.
   • Degree $=$ the power of the highest-order derivative, but only after the equation is expressed as a polynomial in all the derivatives. If the derivative appears inside a transcendental function ($e^{(\cdot)}$, $\sin$, $\log$, etc.), no such polynomial form exists and the degree is undefined.

   Solution …

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6. GUJCET 2019Set 171 markMCQ
   Q.If the general solution of some differential equation is $y = a_1(a_2 + a_3)\cdot\cos(x + a_4) - a_5\,e^{x+a_6}$ then order of differential equation is $\underline{\hspace{2cm}}$. (A) $5$ (B) $4$ (C) $6$ (D) $3$

   ›Reveal solutionSolution

   Count the independent arbitrary constants.

   Concept. The number of independent arbitrary constants equals the order of the differential equation.

   Steps.

   • $a_1(a_2+a_3)\cos(x+a_4)$: the product $a_1(a_2+a_3)$ is one constant and $a_4$ another; but $C\cos(x+a_4)=A\cos x+B\sin x$ gives 2 independent constants. …

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7. GSEB Higher Secondary Certificate (HSC) Examination 2026Set ANNUAL1 markMCQ
   Q.The degree of the differential equation $(1+y_1^2)^{3/2} = y_2$ is ____.

   (a) 4

   (b) 3

   (c) 2

   (d) Not defined

   ›Reveal solutionSolution

   Clear the fractional power by squaring, then read off the power of the highest-order derivative.

   $(1+y_1^2)^{3/2}=y_2$. Squaring both sides to remove the fractional exponent: $(1+y_1^2)^3=y_2^2$.

   …

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8. GSEB Higher Secondary Certificate (HSC) Examination 2025Set ANNUAL1 markMCQ
   Q.The order of the differential equation $\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \cos\left(\dfrac{dy}{dx}\right) + 1 = 0$ = ____.

   (a) $3$

   (b) $2$

   (c) $1$

   (d) Not defined

   ›Reveal solutionSolution

   Order is simply the highest derivative appearing, regardless of the (undefined) degree caused by the $\cos$ term.

   The equation contains $\dfrac{d^2y}{dx^2}$ (raised to a power, inside no transcendental function) and $\dfrac{dy}{dx}$ (which appears inside $\cos(\cdot)$, making the degree undefined -- but order is unaf …

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9. GSEB Higher Secondary Certificate (HSC) Examination 2024Set ANNUAL1 markMCQ
   Q.The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4 + \left(\frac{d^2y}{dx^2}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0$ is ______.

   (a) $3$

   (b) $4$

   (c) $2$

   (d) undefined

   ›Reveal solutionSolution

   The order of a differential equation is the order of the highest derivative present, regardless of the power it's raised to or functions applied to lower-order terms.

   …

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10. GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQ
    Q.The order and degree of the differential equation $1 + \left(\frac{dy}{dx}\right)^2 = \sqrt{\frac{d^2y}{dx^2}}$ respectively are ___.

    (a) $1, 2$

    (b) $2, 2$

    (c) $2, 1$

    (d) $4, 2$

    ›Reveal solutionSolution

    Rationalise, then read off the order (highest derivative) and degree (its power).

    $1+\left(\dfrac{dy}{dx}\right)^2 = \sqrt{\dfrac{d^2y}{dx^2}}.$

    Squaring: $\left(1+\left(\dfrac{dy}{dx}\right)^2\right)^2 = \dfrac{d^2y}{dx^2}.$

    …

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11. GSEB Higher Secondary Certificate (HSC) Examination 2020Set ANNUAL1 markMCQ
    Q.The order and degree of the differential equation $(y''')^3+(y'')^4+(y')^4+y=7$ are ___ respectively.

    (a) 3 and 3

    (b) 1 and 4

    (c) 4 and 1

    (d) 2 and 4

    ›Reveal solutionSolution

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

    $(y''')^3+(y'')^4+(y')^4+y=7$. The highest-order derivative present is $y'''$ (third derivative), so order $=3$.

    …

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12. GSEB Higher Secondary Certificate (HSC) Examination 2019Set ANNUAL1 markMCQ
    Q.The order and degree of $\frac{d^2y}{dx^2} = \sqrt[3]{1+\left(\frac{dy}{dx}\right)^2}$ are ______ respectively.

    (a) $2, 3$

    (b) $3, 2$

    (c) $3$, not defined

    (d) $2, 2$

    ›Reveal solutionSolution

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

    The highest derivative is $\frac{d^2y}{dx^2}$, so the order is $2$. To find the degree, remove the fractional power by cubing both sides: $\left(\frac{d^2y}{dx^2}\right)^3 = 1+\left(\frac{dy}{dx}\right)^2$. Now the equation is a polynomi …

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