Differential Equations · Mathematics · JEE Advanced
MCQ (Single Correct Answer)
1
Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and
$$ \lim\limits_{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1 $$
for each $x>0$. Then, for all $x>0, f(x)$ is equal to :
JEE Advanced 2024 Paper 1 Online
2
Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $3 \int\limits_1^x f(t) d t=x f(x)-\frac{x^3}{3}, x \in[1, \infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is :
JEE Advanced 2023 Paper 2 Online
3
If y = y(x) satisfies the differential equation
$${8\sqrt x \left( {\sqrt {9 + \sqrt x } } \right)dy = {{\left( {\sqrt {4 + \sqrt {9 + \sqrt x } } } \right)}^{ - 1}}}$$
dx, x > 0 and y(0) = $$\sqrt 7 $$, then y(256) =
$${8\sqrt x \left( {\sqrt {9 + \sqrt x } } \right)dy = {{\left( {\sqrt {4 + \sqrt {9 + \sqrt x } } } \right)}^{ - 1}}}$$
dx, x > 0 and y(0) = $$\sqrt 7 $$, then y(256) =
JEE Advanced 2017 Paper 2 Offline
4
The function $$y=f(x)$$ is the solution of the differential equation
$${{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - {x^2}} }}\,$$ in $$(-1,1)$$ satisfying $$f(0)=0$$.
Then $$\int\limits_{ - {{\sqrt 3 } \over 2}}^{{{\sqrt 3 } \over 2}} {f\left( x \right)} \,d\left( x \right)$$ is
$${{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - {x^2}} }}\,$$ in $$(-1,1)$$ satisfying $$f(0)=0$$.
Then $$\int\limits_{ - {{\sqrt 3 } \over 2}}^{{{\sqrt 3 } \over 2}} {f\left( x \right)} \,d\left( x \right)$$ is
JEE Advanced 2014 Paper 2 Offline
5
A curve passes through the point $$\left( {1,{\pi \over 6}} \right)$$. Let the slope of
the curve at each point $$(x,y)$$ be $${y \over x} + \sec \left( {{y \over x}} \right),x > 0.$$
Then the equation of the curve is
the curve at each point $$(x,y)$$ be $${y \over x} + \sec \left( {{y \over x}} \right),x > 0.$$
Then the equation of the curve is
JEE Advanced 2013 Paper 1 Offline
6
Match the statements/expressions in Column I with the values given in Column II:
| Column I | Column II | ||
|---|---|---|---|
| (A) | The number of solutions of the equation $$x{e^{\sin x}} - \cos x = 0$$ in the interval $$\left( {0,{\pi \over 2}} \right)$$ | (P) | 1 |
| (B) | Value(s) of $$k$$ for which the planes $$kx + 4y + z = 0,4x + ky + 2z = 0$$ and $$2x + 2y + z = 0$$ intersect in a straight line | (Q) | 2 |
| (C) | Value(s) of $$k$$ for which $$|x - 1| + |x - 2| + |x + 1| + |x + 2| = 4k$$ has integer solution(s) | (R) | 3 |
| (D) | If $$y' = y + 1$$ and $$y(0) = 1$$ then value(s) of $$y(\ln 2)$$ | (S) | 4 |
| (T) | 5 |
IIT-JEE 2009 Paper 2 Offline
7
Match the statements/expressions in Column I with the open intervals in Column II :
| Column I | Column II | ||
|---|---|---|---|
| (A) | Interval contained in the domain of definition of non-zero solutions of the differential equation $${(x - 3)^2}y' + y = 0$$ | (P) | $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ |
| (B) | Interval containing the value of the integral $$\int\limits_1^5 {(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)dx} $$ | (Q) | $$\left( {0,{\pi \over 2}} \right)$$ |
| (C) | Interval in which at least one of the points of local maximum of $${\cos ^2}x + \sin x$$ lies | (R) | $$\left( {{\pi \over 8},{{5\pi } \over 4}} \right)$$ |
| (D) | Interval in which $${\tan ^{ - 1}}(\sin x + \cos x)$$ is increasing | (S) | $$\left( {0,{\pi \over 8}} \right)$$ |
| (T) | $$( - \pi ,\pi )$$ |
IIT-JEE 2009 Paper 1 Offline
8
Let a solution $$y=y(x)$$ of the differential equation,
$$x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0$$ satify $$y\left( 2 \right) = {2 \over {\sqrt 3 }}.$$
$$x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0$$ satify $$y\left( 2 \right) = {2 \over {\sqrt 3 }}.$$
STATEMENT-1 : $$y\left( x \right) = \sec \left( {{{\sec }^{ - 1}}x - {\pi \over 6}} \right)$$ and
STATEMENT-2 : $$y\left( x \right)$$ given by $${1 \over y} = {{2\sqrt 3 } \over x} - \sqrt {1 - {1 \over {{x^2}}}} $$
IIT-JEE 2008 Paper 2 Offline
9
The differential equation $${{dy} \over {dx}} = {{\sqrt {1 - {y^2}} } \over y}$$ determines a family of circles with
IIT-JEE 2005 Screening
10
If $$y=y(x)$$ and it follows the relation $$x\cos \,y + y\,cos\,x = \pi $$ then $$y''(0)=$$
IIT-JEE 2005 Screening
11
For the primitive integral equation $$ydx + {y^2}dy = x\,dy;$$
$$x \in R,\,\,y > 0,y = y\left( x \right),\,y\left( 1 \right) = 1,$$ then $$y(-3)$$ is
$$x \in R,\,\,y > 0,y = y\left( x \right),\,y\left( 1 \right) = 1,$$ then $$y(-3)$$ is
IIT-JEE 2005 Screening
12
The solution of primitive integral equation $$\left( {{x^2} + {y^2}} \right)dy = xy$$
$$dx$$ is $$y=y(x),$$ If $$y(1)=1$$ and $$\left( {{x_0}} \right) = e$$, then $${{x_0}}$$ is equal to
$$dx$$ is $$y=y(x),$$ If $$y(1)=1$$ and $$\left( {{x_0}} \right) = e$$, then $${{x_0}}$$ is equal to
IIT-JEE 2005 Screening
13
If $$y=y(x)$$ and $${{2 + \sin x} \over {y + 1}}\left( {{{dy} \over {dx}}} \right) = - \cos x,y\left( 0 \right) = 1,$$
then $$y\left( {{\pi \over 2}} \right)$$ equals
then $$y\left( {{\pi \over 2}} \right)$$ equals
IIT-JEE 2004 Screening
14
If $$y(t)$$ is a solution of $$\left( {1 + t} \right){{dy} \over {dt}} - ty = 1$$ and $$y\left( 0 \right) = - 1,$$ then $$y(1)$$ is equal to
IIT-JEE 2003 Screening
15
If $${x^2} + {y^2} = 1,$$ then
IIT-JEE 2000 Screening
16
A solution of the differential equation
$${\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0$$ is
$${\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0$$ is
IIT-JEE 1999
17
The order of the differential equation whose general solution is given by
$$y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{e^{x + {C_5}}},$$ where
$${C_1},{C_2},{C_3},{C_4},{C_5},$$ are arbitrary constants, is
$$y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{e^{x + {C_5}}},$$ where
$${C_1},{C_2},{C_3},{C_4},{C_5},$$ are arbitrary constants, is
IIT-JEE 1998
Numerical
1
For $x \in \mathbb{R}$, let $y(x)$ be a solution of the differential equation
$\left(x^2-5\right) \frac{d y}{d x}-2 x y=-2 x\left(x^2-5\right)^2$ such that $y(2)=7$.
Then the maximum value of the function $y(x)$ is :
$\left(x^2-5\right) \frac{d y}{d x}-2 x y=-2 x\left(x^2-5\right)^2$ such that $y(2)=7$.
Then the maximum value of the function $y(x)$ is :
JEE Advanced 2023 Paper 2 Online
2
If $y(x)$ is the solution of the differential equation
$$ x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x > 0, y(1)=2, $$
and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is
$$ x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x > 0, y(1)=2, $$
and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is
JEE Advanced 2022 Paper 2 Online
3
Let f : R $$ \to $$ R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation $${{dy} \over {dx}} = (2 + 5y)(5y - 2)$$, then the value of $$\mathop {\lim }\limits_{n \to - \infty } f(x)$$ is ...........
JEE Advanced 2018 Paper 2 Offline
4
Let $$f:[1,\infty ) \to [2,\infty )$$ be a differentiable function such that $$f(1) = 2$$. If $$6\int\limits_1^x {f(t)dt = 3xf(x) - {x^3} - 5} $$ for all $$x \ge 1$$, then the value of f(2) is ___________.
IIT-JEE 2011 Paper 1 Offline
5
Let $$y'\left( x \right) + y\left( x \right)g'\left( x \right) = g\left( x \right),g'\left( x \right),y\left( 0 \right) = 0,x \in R,$$ where $$f'(x)$$ denotes $${{df\left( x \right)} \over {dx}}$$ and $$g(x)$$ is a given non-constant differentiable function on $$R$$ with $$g(0)=g(2)=0.$$ Then the value of $$y(2)$$ is
IIT-JEE 2011 Paper 2 Offline
MCQ (More than One Correct Answer)
1
For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation
$$ \frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0 $$
Then, which of the following statements is/are TRUE ?
$$ \frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), \quad y(0)=0 $$
Then, which of the following statements is/are TRUE ?
JEE Advanced 2022 Paper 2 Online
2
Let $$\Gamma $$ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the y-axis at YP. If PYP has length 1 for each point P on I`, then which of the following options is/are correct?
JEE Advanced 2019 Paper 1 Offline
3
If $$g(x) = \int_{\sin x}^{\sin (2x)} {{{\sin }^{ - 1}}} (t)\,dt$$, then
JEE Advanced 2017 Paper 2 Offline
4
A solution curve of the differential equation
$$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$$ $$x>0,$$ passes through the
point $$(1,3)$$. Then the solution curve
$$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$$ $$x>0,$$ passes through the
point $$(1,3)$$. Then the solution curve
JEE Advanced 2016 Paper 1 Offline
5
Let $$f:(0,\infty ) \to R$$ be a differentiable function such that $$f'(x) = 2 - {{f(x)} \over x}$$ for all $$x \in (0,\infty )$$ and $$f(1) \ne 1$$. Then
JEE Advanced 2016 Paper 1 Offline
6
Consider the family of all circles whose centres lie on the straight line $$y=x,$$ If this family of circle is represented by the differential equation $$Py'' + Qy' + 1 = 0,$$ where $$P, Q$$ are functions of $$x,y$$ and $$y'$$ $$\left( {here\,\,\,y' = {{dy} \over {dx}},y'' = {{{d^2}y} \over {d{x^2}}}} \right)$$ then which of the following statements is (are) true?
JEE Advanced 2015 Paper 1 Offline
7
Let $$y(x)$$ be a solution of the differential equation
$$\left( {1 + {e^x}} \right)y' + y{e^x} = 1.$$
If $$y(0)=2$$, then which of the following statement is (are) true?
$$\left( {1 + {e^x}} \right)y' + y{e^x} = 1.$$
If $$y(0)=2$$, then which of the following statement is (are) true?
JEE Advanced 2015 Paper 1 Offline
8
If $$y(x)$$ satisfies the differential equation $$y' - y\,tan\,x = 2x\,secx$$ and $$y(0)=0,$$ then
IIT-JEE 2012 Paper 1 Offline
9
A curve $$y=f(x)$$ passes through $$(1,1)$$ and at $$P(x,y),$$ tangent cuts the $$x$$-axis and $$y$$-axis at $$A$$ and $$B$$ respectively such that $$BP:AP=3:1,$$ then
IIT-JEE 2006
10
The differential equation representing the family of curves
$${y^2} = 2c\left( {x + \sqrt c } \right),$$ where $$c$$ is a positive parameter, is of
$${y^2} = 2c\left( {x + \sqrt c } \right),$$ where $$c$$ is a positive parameter, is of
IIT-JEE 1999
Subjective
1
If length of tangent at any point on the curve $$y=f(x)$$ intecepted between the point and the $$x$$-axis is length $$1.$$ Find the equation of the curve.
IIT-JEE 2005
2
A curve $$'C''$$ passes through $$(2,0)$$ and the slope at $$(x,y|)$$ as $$\,{{{{\left( {x + 1} \right)}^2} + \left( {y - 3} \right)} \over {x + 3}}$$. Find the equation of the curve. Find the area bounded by curve and $$x$$-axis in fourth quadrant.
IIT-JEE 2004
3
A right circular cone with radius $$R$$ and height $$H$$ contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant $$ = k > 0$$. Find the time after which the come is empty.
IIT-JEE 2003
4
A hemispherical tank of radius $$2$$ metres is initially full of water and has an outlet of $$12$$ cm2 cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law $$v(t)=0.6$$ $$\sqrt {2gh\left( t \right),} $$ where $$v(t)$$ and $$h(t)$$ are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time $$t,$$ and $$g$$ is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint: From a differential equation by relasing the decreases of water level to the outflow).
IIT-JEE 2001
5
Let $$u(x)$$ and $$v(x)$$ satisfy the differential equation $${{du} \over {dx}} + p\left( x \right)u = f\left( x \right)$$ and $${{dv} \over {dx}} + p\left( x \right)v = g\left( x \right),$$ where $$p(x) f(x)$$ and $$g(x)$$ are continuous functions. If $$u\left( {{x_1}} \right) > v\left( {{x_1}} \right)$$ for some $${{x_1}}$$ and $$f(x)>g(x)$$ for all $$x > {x_1},$$ prove that any point $$(x,y)$$ where $$x > {x_1},$$ does not satisfy the equations $$y=u(x)$$ and $$y=v(x)$$
IIT-JEE 1997
6
Determine the equation of the curve passing through the origin, in the form $$y=f(x),$$ which satisfies the differential equation $${{dy} \over {dx}} = \sin \left( {10x + 6y} \right).\,$$
IIT-JEE 1996
7
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 $$2.$$ From the differential equation and determine all such possible curves.
IIT-JEE 1995
8
A normal is drawn at a point $$P(x,y)$$ of a curve. It meets the $$x$$-axis at $$Q.$$ If $$PQ$$ is of constant length $$k,$$ then show that the differential equation describing such curves is $$y = {{dy} \over {dx}} = \pm \sqrt {{k^2} - {y^2}} $$
Find the equation of such a curve passing through $$(0,k).$$
IIT-JEE 1994
9
If $$\left( {a + bx} \right){e^{y/x}} = x,$$ then prove that $${x^3}{{{d^2}y} \over {d{x^2}}} = {\left( {x{{dy} \over {dx}} - y} \right)^2}$$
IIT-JEE 1983