## MCQ (Single Correct Answer)

The degree of the differential equation $${\left[ {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} \right]^{5/3}} = {{{d^2}y} \over {d{x^2}}}$$ is...

The differential equation of all parabolas whose axes are parallel to y-axis is

The solution of the differential equation $${{dy} \over {dx}} = {e^{y + x}} + {e^{y - x}}$$ is

The order and degree of the following differential equation $${\left[ {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} \right]^{5/2}} = {{{d^3}y} \over ...

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

The differential equation of the family of curves $$y = {e^{2x}}(a\cos x + b\sin x)$$, where a and b are arbitrary constants, is given by

The slope at any point of a curve y = f(x) is given by $${{dy} \over {dx}} = 3{x^2}$$ and it passes through ($$-$$1, 1). The equation of the curve is...

The general solution of the differential equation $${{dy} \over {dx}} = {e^{y + x}} + {e^{y - x}}$$ is
where c is an arbitrary constant

The integrating factor of the differential equation $$x\log x{{dy} \over {dx}} + y = 2\log x$$ is given by

If x2 + y2 = 1, then

The order of the differential equation $${{{d^2}y} \over {d{x^2}}} = \sqrt {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} $$ is

The general solution of the differential equation $$100{{{d^2}y} \over {d{x^2}}} - 20{{dy} \over {dx}} + y = 0$$ is

If $$y'' - 3y' + 2y = 0$$ where y(0) = 1, y'(0) = 0, then the value of y at $$x = {\log _e}2$$ is

The degree of the differential equation $$x = 1 + \left( {{{dy} \over {dx}}} \right) + {1 \over {2!}}{\left( {{{dy} \over {dx}}} \right)^2} + {1 \over...

The equation of one of the curves whose slope at any point is equal to y + 2x is

Solution of the differential equation xdy $$-$$ ydx = 0 represents a

If the displacement, velocity and acceleration of a particle at time t be x, v and f respectively, then which one is true?

The displacement x of a particle at time t is given by x = At2 + Bt + C, where A, B, C are constants and v is velocity of a particle, then the value o...

The displacement of a particle at time t is x, where x = t4 $$-$$ kt3. If the velocity of the particle at time t = 2 is minimum, then...

The general solution of the differential equation $${{{d^2}y} \over {d{x^2}}} + 8{{dy} \over {dx}} + 16y = 0$$ is

The degree and order of the differential equation $$y = x{\left( {{{dy} \over {dx}}} \right)^2} + {\left( {{{dx} \over {dy}}} \right)^2}$$ are respect...

he general solution of the differential equation $${\log _e}\left( {{{dy} \over {dx}}} \right) = x + y$$ is

The solution of $${{dy} \over {dx}} = {y \over x} + \tan {y \over x}$$ is

Integrating factor (I.F.) of the differential equation $${{dy} \over {dx}} - {{3{x^2}} \over {1 + {x^3}}}y = {{{{\sin }^2}x} \over {1 + x}}$$ is...

The differential equation of y = aebx (a & b are parameters) is

The solution of $$\cos y{{dy} \over {dx}} = {e^{x + \sin y}} + {x^2}{e^{\sin y}}$$ is $$f(x) + {e^{ - \sin y}} = C$$ (C is arbitrary real constant) wh...

If the transformation $$z = \log \tan {x \over 2}$$ reduces the differential equation $${{{d^2}y} \over {d{x^2}}} + \cot x{{dy} \over {dx}} + 4y\cos e...

The differential equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is where $$y^{\prime}\equiv{{{dx}\over {dy}}...

If $$x{{dy} \over {dx}} + y = {{xf(xy)} \over {f'(xy)'}}$$, then | f(xy) | is equal to (where k is an arbitrary positive constant).

The differential of $$f(x) = {\log _e}(1 + {e^{10x}}) - {\tan ^{ - 1}}({e^{5x}})$$ at x = 0 and for dx = 0.2 is

Let cos$$^{ - 1}\left( {{y \over b}} \right) = \log {\left( {{x \over n}} \right)^n}$$. Then

Let f be a differentiable function with $$\mathop {\lim }\limits_{x \to \infty } f(x) = 0.$$ If $$y' + yf'(x) - f(x)f'(x) = 0$$, $$\mathop {\lim }\lim...

If $$x\sin \left( {{y \over x}} \right)dy = \left[ {y\sin \left( {{y \over x}} \right) - x} \right]dx,\,x > 0$$ and $$y(1) = {\pi \over 2}$$, then...

The differential equation of the family of curves y = ex (A cos x + B sin x) where, A, B are arbitrary constants is

Let $$y = f(x) = 2{x^2} - 3x + 2$$. The differential of y when x changes from 2 to 1.99 is

Let $$y = {1 \over {1 + x + lnx}}$$, then

The general solution of the differential equation $$\left( {1 + {e^{{x \over y}}}} \right)dx + \left( {1 - {x \over y}} \right){e^{x/y}}dy = 0$$ is (C...

General solution of $${(x + y)^2}{{dy} \over {dx}} = {a^2},a \ne 0$$ is (C is an arbitrary constant)

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

Let y(x) be a solution of $$(1 + {x^2}){{dy} \over {dx}} + 2xy - 4{x^2} = 0$$. Then y(1) is equal to

If $$y = {e^{m{{\sin }^{ - 1}}x}}$$ then $$(1 - {x^2}){{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} - $$ky = 0, where k is equal to

Solution of $${(x + y)^2}{{dy} \over {dx}} = {a^2}$$ ('a' belong a constant) is

The integrating factor of the first order differential equation $${x^2}({x^2} - 1){{dy} \over {dx}} + x({x^2} + 1)y = {x^2} - 1$$ is

## Subjective

Show that sin x is a monotonic increasing function of x in 0

If x = sin t, y = sin 2t, prove that $$(1 - {x^2}){{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} + 4y = 0$$

If f is differentiable at x = a, find the value of $$\mathop {\lim }\limits_{x \to a} {{{x^2}f(a) - {a^2}f(x)} \over {x - a}}$$

If $${{dy} \over {dx}} + \sqrt {{{1 - {y^2}} \over {1 - {x^2}}}} = 0$$, prove that $$x\sqrt {1 - {y^2}} + y\sqrt {1 - {x^2}} = A$$, where A is a co...

Let R be the set of real numbers and f : R $$\to$$ R be such that for all x, y $$\in$$ R, $$|f(x) - f(y)| \le |x - y{|^3}$$. Prove that f is a constan...

Find the general solution of $$(x + \log y)dy + y\,dx = 0$$