Formation of Differential Equations

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JEE Main 2009

The differential equation which represents the family of curves $y = c_{1} e^{c_{2} x}$ , where $c_{1}$ and $c_{2}$ are arbitrary constants, is

y^{''} = y^{'} y

(B)

y y^{''} = y^{'}

(C)

y y^{''} = (y^{'})^{2}

(D)

y^{'} = y^{2}

JEE Main 2008

LEVELJEE Main

The differential equation of the family of circles with fixed radius 5 units and centre on the line $y = 2$ is

(A)

(x - 2) y^{'2} = 25 - (y - 2)^{2}

(B)

(y - 2) y^{'2} = 25 - (y - 2)^{2}

(C)

(y - 2)^{2} y^{'2} = 25 - (y - 2)^{2}

(D)

(x - 2)^{2} y^{'2} = 25 - (y - 2)^{2}

JEE Main 2007

LEVELJEE Main

The differential equation of all circles passing through the origin and having their centres on the $x$ -axis is

(A)

y^{2} = x^{2} + 2 x y \frac{d y}{d x}

(B)

y^{2} = x^{2} - 2 x y \frac{d y}{d x}

(C)

x^{2} = y^{2} + x y \frac{d y}{d x}

(D)

x^{2} = y^{2} + 3 x y \frac{d y}{d x}

JEE Main 2006

LEVELJEE Main

The differential equation whose solution is $A x^{2} + B y^{2} = 1$ where $A$ and $B$ are arbitrary constants is of

(A)

second order and second degree

(B)

first order and second degree

(C)

first order and first degree

(D)

second order and first degree

JEE Main 2004

LEVELJEE Main

The differential equation for the family of circle $x^{2} + y^{2} - 2 a y = 0$ , where $a$ is an arbitrary constant is

(A)

(x^{2} + y^{2}) y^{'} = 2 x y

(B)

2 (x^{2} + y^{2}) y^{'} = x y

(C)

(x^{2} - y^{2}) y^{'} = 2 x y

(D)

2 (x^{2} - y^{2}) y^{'} = x y

JEE Advanced 2015

LEVELJEE Main

Consider the family of all circles whose centers lie on the straight line $y = x$ . If this family of circle is represented by the differential equation $P y^{''} + Q y^{'} + 1 = 0$ , where $P, Q$ are functions of $x, y$ and $y^{'}$ , then which of the following statements is (are) true?

* Multiple Correct Options

P + Q = 1 - x + y + y' + (y')^2

(D)

P - Q = x + y - y' - (y')^2

JEE Advanced 1999

LEVELJEE Main

The differential equation representing the family of curves $y^{2} = 2 c (x + c)$ , where $c$ is a positive parameter, is of

* Multiple Correct Options

Let $y = f (x)$ be a curve passing through (1, 1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Form the differential equation and determine all such possible curves.

JEE Main 2020 (8 January Shift 2)

LEVELJEE Main

The differential equation of the family of curves, $x^{2} = 4 b (y + b)$ , $b \in R$ , is:

(A)

x y^{''} = y^{'}

(B)

x (y^{'})^{2} = x + 2 y y^{'}

(C)

x (y^{'})^{2} = x - 2 y y^{'}

(D)

x (y^{'})^{2} = 2 y y^{'} - x

JEE Main 2020 - 8 Jan (Evening)

LEVELJEE Main

The differential equation of the family of curves, $x^{2} = 4 b (y + b)$ , $b \in R$ , is

(A)

x (\frac{d y}{d x})^{2} = 2 y \frac{d y}{d x} + x^{2}

(B)

x (\frac{d y}{d x})^{2} = 2 y \frac{d y}{d x} + x

(C)

x (\frac{d y}{d x})^{2} = y \frac{d y}{d x} + x^{2}

(D)

x (\frac{d y}{d x})^{2} = y \frac{d y}{d x} + 2 x^{2}

JEE Main 2021 (27 August Shift 2)

LEVELJEE Main

A differential equation representing the family of parabolas with axis parallel to $y$ -axis and whose length of latus rectum is the distance of the point $(2, - 3)$ from the line $3 x + 4 y = 5$ , is given by :

(A)

10 \frac{d ^{2} y}{d x ^{2}} = 11

(B)

11 \frac{d ^{2} x}{d y ^{2}} = 10

(C)

10 \frac{d ^{2} x}{d y ^{2}} = 11

(D)

11 \frac{d ^{2} y}{d x ^{2}} = 10

JEE Main 2021 (18 March Shift 1)

LEVELJEE Main

The differential equation satisfied by the system of parabolas $y^{2} = 4 a (x + a)$ is:

(A)

y (\frac{d y}{d x})^{2} - 2 x (\frac{d y}{d x}) - y = 0

(B)

y (\frac{d y}{d x})^{2} - 2 x (\frac{d y}{d x}) + y = 0

(C)

y (\frac{d y}{d x})^{2} + 2 x (\frac{d y}{d x}) - y = 0

(D)

y (\frac{d y}{d x}) + 2 x (\frac{d y}{d x}) - y = 0

JEE Main 2022 (28 July Shift 2)

LEVELJEE Main

The differential equation of the family of circles passing through the points $(0, 2)$ and $(0, - 2)$ is

(A)

2 x y \frac{d y}{d x} + (x^{2} - y^{2} + 4) = 0

(B)

2 x y \frac{d y}{d x} + (x^{2} + y^{2} - 4) = 0

(C)

2 x y \frac{d y}{d x} + (y^{2} - x^{2} + 4) = 0

(D)

2 x y \frac{d y}{d x} - (x^{2} - y^{2} + 4) = 0

JEE Main 2024 (08 Apr Shift 1)

LEVELJEE Main

Let $f (x)$ be a positive function such that the area bounded by $y = f (x), y = 0$ from $x = 0$ to $x = a > 0$ is $e^{- a} + 4 a^{2} + a - 1$ . Then the differential equation, whose general solution is $y = c_{1} f (x) + c_{2}$ , where $c_{1}$ and $c_{2}$ are arbitrary constants, is

(A)

(8 e^{x} - 1) \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = 0

(B)

(8 e^{x} - 1) \frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} = 0

(C)

(8 e^{x} + 1) \frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} = 0

(D)

(8 e^{x} + 1) \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = 0

JEE Main 2024 (05 April Shift 2)

LEVELJEE Main

The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :

(A)

(x^{2} - y^{2} + 2 x y) d x = (x^{2} - y^{2} - 2 x y) d y

(B)

(x^{2} + y^{2} + 2 x y) d x = (x^{2} + y^{2} - 2 x y) d y

(C)

(x^{2} + y^{2} - 2 x y) d x = (x^{2} + y^{2} + 2 x y) d y

(D)

(x^{2} - y^{2} + 2 x y) d x = (x^{2} - y^{2} + 2 x y) d y

JEE Advanced 2018

LEVELJEE Main

Let $f : R \to R$ be a differentiable function with $f (0) = 1$ and satisfying the equation $f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y)$ for all $x, y \in R$ . Then, the value of $lo g_{e} (f (4))$ is ________.

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f′(y)+f′(x)f(y) for all x,y∈R. Then, the value of loge​(f(4)) is ________.

The differential equation which represents the family of curves $y = c_{1} e^{c_{2} x}$ , where $c_{1}$ and $c_{2}$ are arbitrary constants, is

The differential equation of the family of circles with fixed radius 5 units and centre on the line $y = 2$ is

The differential equation of all circles passing through the origin and having their centres on the $x$ -axis is

The differential equation whose solution is $A x^{2} + B y^{2} = 1$ where $A$ and $B$ are arbitrary constants is of

The differential equation for the family of circle $x^{2} + y^{2} - 2 a y = 0$ , where $a$ is an arbitrary constant is

The differential equation representing the family of curves $y^{2} = 2 c (x + c)$ , where $c$ is a positive parameter, is of

The differential equation of the family of curves, $x^{2} = 4 b (y + b)$ , $b \in R$ , is:

The differential equation of the family of curves, $x^{2} = 4 b (y + b)$ , $b \in R$ , is

The differential equation satisfied by the system of parabolas $y^{2} = 4 a (x + a)$ is:

The differential equation of the family of circles passing through the points $(0, 2)$ and $(0, - 2)$ is

The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :

Let $f : R \to R$ be a differentiable function with $f (0) = 1$ and satisfying the equation $f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y)$ for all $x, y \in R$ . Then, the value of $lo g_{e} (f (4))$ is ________.