Q.(vi) The differential equation representing the family of circles $x^2+(y-a)^2=a^2$ will be of order two. (State True or False.)

Showing the 12 most recent of 13 on this concept.

1. 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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2. 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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3. 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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4. 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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5. 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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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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