## Numerical

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 c...

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 - ...

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 ...

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 $$f'(x)$$...

## MCQ (More than One Correct Answer)

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\righ...

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 intersec...

If $$g(x) = \int_{\sin x}^{\sin (2x)} {{{\sin }^{ - 1}}} (t)\,dt$$, then

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 ...

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 $$f(1) \ne 1$$. ...

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 sta...

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 equatio...

If $$y(x)$$ satisfies the differential equation $$y' - y\,tan\,x = 2x\,secx$$ and $$y(0)=0,$$ then

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 $...

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

## MCQ (Single Correct Answer)

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 } } } \r...

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 - ...

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 \l...

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 \ov...

If $$y=y(x)$$ and it follows the relation $$x\cos \,y + y\,cos\,x = \pi $$ then $$y''(0)=$$

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 $$y(-3)$$...

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)...

The differential equation $${{dy} \over {dx}} = {{\sqrt {1 - {y^2}} } \over y}$$ determines a family of circles with

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 ...

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

If $${x^2} + {y^2} = 1,$$ then

A solution of the differential equation
$${\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0$$ is

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}{...

## Subjective

Match the statements/expressions in Column $$I$$ with the open intervals in Column $$II$$.
Column $$I$$
(A)$$\,\,\,$$ Interval contained in the domai...

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 cu...

A curve $$'C''$$ passes through $$(2,0)$$ and the slope at $$(x,y|)$$ as $$\,{{{{\left( {x + 1} \right)}^2} + \left( {y - 3} \right)} \over {x + 3}}$$...

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...

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 ...

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...

Determine the equation of the curve passing through the origin, in the form $$y=f(x),$$ which satisfies the differential equation $${{dy} \over {dx}} ...

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 lie...

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 differ...

If $$\left( {a + bx} \right){e^{y/x}} = x,$$ then prove that $${x^3}{{{d^2}y} \over {d{x^2}}} = {\left( {x{{dy} \over {dx}} - y} \right)^2}$$