Family y=Ax+A³ of curves is represented by the differential equation of degree: (A) 1 (B) 2 (C) 3 (D) 4

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y=Ax+A^3

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Eliminating $A$ gives $y=x\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^3$ , in which the highest-order derivative $\frac{dy}{dx}$ occurs to the power 3, so the degree is 3 — option (C).

Find the differential equation

The family $y=Ax+A^3$ has one arbitrary constant $A$ . Differentiate once:

\frac{dy}{dx}=A.

So $A=\frac{dy}{dx}$ . Substitute this back into $y=Ax+A^3$ :

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

Read off the degree …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

KEAM 2026Set eng-2026-04224 marksMCQ
$2\frac{dy}{dx} - 3x = \left(2y - x\frac{dy}{dx}\right)^{-3}$
›Reveal solutionSolution

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

Given $2\frac{dy}{dx}-3x=\Big(2y-x\frac{dy}{dx}\Big)^{-3}$ .

Multiply through by $\big(2y-x\,y'\big)^3$ : $\big(2y'-3x\big)\big(2y-x\,y'\big)^3=1$ .

The highest-order derivative is $y'$ , so the order is $1$ . …
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KEAM 2024Set eng-2024-06094 marksMCQ
$(y''')^2 + (\sin y')^4 + y = 0$
›Reveal solutionSolution

Degree requires a polynomial form in the derivatives; the $(\sin y')^4$ term makes that impossible, so the degree is not defined.

The degree of a differential equation is defined only when it is a polynomial in all its derivatives. …
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KEAM 2026Set eng-2026-04204 marksMCQ
$\left(1 + \frac{dy}{dx} + \frac{d^2y}{dx^2}\right)^{\frac{3}{2}} = \left(x + y + \frac{dy}{dx} + \frac{d^3y}{dx^3}\right)^{\frac{2}{3}}$
›Reveal solutionSolution

Order = highest derivative; degree = its power after removing fractional exponents.

The highest derivative present is $\dfrac{d^3y}{dx^3}$ , so the order is $3$ . …
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KEAM 2025Set eng-2025-04264 marksMCQ
$c_1, c_2, c_3$
›Reveal solutionSolution

Count the number of essential (independent) arbitrary constants.

Write $A=c_1+c_2$ (a single constant). Then

$y=A\sin(x+c_3)-c_4e^x=(A\cos c_3)\sin x+(A\sin c_3)\cos x-c_4e^x.$ …
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KEAM 2024Set eng-2024-06084 marksMCQ
$\dfrac{d^2y}{dx^2}-2x=\sqrt{y+\dfrac{dy}{dx}}$
›Reveal solutionSolution

Order $=2$ from $\dfrac{d^2y}{dx^2}$ ; rationalizing the square root raises this second derivative to power $2$ , so degree $=2$ .

The equation is
$\frac{d^2y}{dx^2}-2x=\sqrt{\,y+\frac{dy}{dx}\,}.$
Order: the highest-order derivative present is $\dfrac{d^2y}{dx^2}$ , so the order is $2$ .

Degree: degree is the power of the highest-order derivative after the equation is made polynomial (free of radicals) in its derivatives. Squaring both sides: …
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KEAM 2025Set eng-2025-04274 marksMCQ
$c_1, c_2, c_3, c_4$
›Reveal solutionSolution
Although four letters appear, $c_1$ and $c_2$ only occur as the sum $c_1+c_2$ , so there are effectively $3$ independent arbitrary constants and the differential equation has order $3$ .

In $y=(c_1+c_2)\sin(2x+c_3)+c_4 e^{5x}$ , the constants $c_1$ and $c_2$ appear solely through their sum $A=c_1+c_2$ , which behaves as one arbitrary constant.

Thus the genuinely independent arbitrary constants are:

the amplitude $A=c_1+c_2$ ,

the phase $c_3$ , …
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KEAM 2025Set eng-2025-04294 marksMCQ
$\sqrt[5]{1 + \dfrac{d^2 y}{dx^2}} = \sqrt[4]{\left(y + \left(\dfrac{dy}{dx}\right)^5\right)}$
›Reveal solutionSolution

Order $=2$ (from $y''$ ); after clearing the radicals the power of $y''$ is $4$ , so degree $=4$ . Answer $(2,4)$ .

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

Degree. The equation is
$\left(1+\frac{d^2y}{dx^2}\right)^{1/5} = \left(y+\left(\frac{dy}{dx}\right)^5\right)^{1/4}.$
Degree requires the equation to be a polynomial in the derivatives. Raise both sides to the power $20$ (LCM of $5$ and $4$ ): …
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Q.Family $y=Ax+A^3$ of curves is represented by the differential equation of degree:
(A) 1
(B) 2
(C) 3
(D) 4

Concept understanding — Order Of Differential Equation

Order of a Differential Equation

Some examples

Why order matters …

Find the differential equation

Read off the degree …

Method: Degree of the DE of a Family Given by a Parameter

Steps

Common Mistakes

Q.Family y=Ax+A3y=Ax+A^3y=Ax+A3 of curves is represented by the differential equation of degree: (A) 1 (B) 2 (C) 3 (D) 4

Find the differential equation

Read off the degree …

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Q.Family $y=Ax+A^3$ of curves is represented by the differential equation of degree:
(A) 1
(B) 2
(C) 3
(D) 4

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