Differential Equations · Mathematics · WB JEE

The function f(x) which satisfies $$f(x) = f( - x) = {{f'(x)} \over x}$$ is given by

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 $${\left[ {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} \right]^{5/2}} = {{{d^3}y} \over {d{x^3}}}$$ are respectively

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 $$x = 1 + \left( {{{dy} \over {dx}}} \right) + {1 \over {2!}}{\left( {{{dy} \over {dx}}} \right)^2} + {1 \over {3!}}{\left( {{{dy} \over {dx}}} \right)^3} + .....$$

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 of 4Ax $$-$$ v² is

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 respectively

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 = ae^bx (a & b are parameters) is

Let $$\mathrm{f}$$ be a differential function with $$\lim _\limits{x \rightarrow \infty} \mathrm{f}(x)=0$$. If $$\mathrm{y}^{\prime}+\mathrm{yf}^{\prime}(x)-\mathrm{f}(x) \mathrm{f}^{\prime}(x)=0$$, $$\lim _\limits{x \rightarrow \infty} y(x)=0$$ then

If $$x y^{\prime}+y-e^x=0, y(a)=b$$, then $$\lim _\limits{x \rightarrow 1} y(x)$$ is

If $$y = {x \over {{{\log }_e}|cx|}}$$ is the solution of the differential equation $${{dy} \over {dx}} = {y \over x} + \phi \left( {{x \over y}} \right)$$, then $$\phi \left( {{x \over y}} \right)$$ is given by

Given $${{{d^2}y} \over {d{x^2}}} + \cot x{{dy} \over {dx}} + 4y\cos e{c^2}x = 0$$. Changing the independent variable x to z by the substitution $$z = \log \tan {x \over 2}$$, the equation is changed to

The family of curves $$y = {e^{a\sin x}}$$, where 'a' is arbitrary constant, is represented by the differential equation

If $$x{{dy} \over {dx}} + y = x{{f(xy)} \over {f'(xy)}}$$, then $$|f(xy)|$$ is equal to

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) where f(x) is equal to

If the transformation $$z = \log \tan {x \over 2}$$ reduces the differential equation

$${{{d^2}y} \over {d{x^2}}} + \cot x{{dy} \over {dx}} + 4y\cos e{c^2}x = 0$$ into the form $${{{d^2}y} \over {d{z^2}}} + ky = 0$$ then k is equal to

Solve : $$({x^2} + 4{y^2} + 4xy)dy = (2x + 4y + 1)dx$$

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 constant.

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