Differential Equations

Let $f:(0,1) \rightarrow(0,1)$ be a differentiable function such that $f^{\prime}(x) \neq 0 \forall x \in(0,1)$ and $f\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$. Suppose for all $x$, $\mathop {\lim }\limits_{t \to x} \frac{\int_0^t \sqrt...

The solution of the differential equation $2 x^2 y \frac{d y}{d x}=\tan \left(x^2 y^2\right)-2 x y^2$, given $y(1)=\sqrt{\frac{\pi}{2}}$ is

The population $p(t)$ at time $t$ of a certain mouse species follows the differential equation
\(\frac{d p(t)}{d t}=0.5 p(t)-450\)
If $p(0)=850$, then the time at which the population becomes zero is

If $x=\int\limits_0^y \frac{1}{\sqrt{1+9 t^2}} d t$ and $\frac{d^2 y}{d x^2}=a y$, then $a$ is equal to

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

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

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

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

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

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

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

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 equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is where \(y^{\prime}\equiv{{{dx}\over {dy}}},y^{\prime\prime}\equiv{{{d^2}y\over {dx^2}}}\)

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

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

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

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

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 value of \(\cos \left( {{y \over x}} \right)\) is

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

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 is an arbitrary constant)

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

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

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

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

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

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

General solution of \(y{{dy} \over {dx}} + b{y^2} = a\cos x,0 < x < 1\) is

If the solution of the differential equation \(x{{dy} \over {dx}} + y = x{e^x}\,be\,xy = {e^x}\phi (x) + C\), then \(\phi\)(x) is equal to

The general solution of the differential equation \({{dy} \over {dx}} = {{x + y + 1} \over {2x + 2y + 1}}\) is

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

The differential equation of y = aebx (a & b are parameters) 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 solution of \({{dy} \over {dx}} = {y \over x} + \tan {y \over x}\) is

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

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

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.

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 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

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

Solution of the differential equation xdy \(-\) ydx = 0 represents a

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

The degree of the differential equation \(x = 1 + \left( {{{dy} \over {dx}}} \right) + {1 \over {2!}}{\left( {{{dy} \over {dx}}} \right)^2} + {1 \over {3!}}{\left( {{{dy} \over {dx}}} \right)^3} + .....\)

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

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

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

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