Differential Equations

If a curve y = f(x), passing through the point
(1, 2), is the solution of the differential equation,
2x2dy= (2xy + y2)dx, then \(f\left( {{1 \over 2}} \right)\) is equal to :

If y = y(x) is the solution of the differential equation,
x\(dy \over dx\) + 2y = x2, satisfying y(1) = 1, then y(\(1\over2\)) is equal
to :

Let f : [0,1] \(\to\) R be such that f(xy) = f(x).f(y), for all x, y \(\in\) [0, 1], and f(0) \(\ne\) 0. If y = y(x) satiesfies the differential equation, \({{dy} \over {dx}}\) = f(x) with y(0) = 1, then y$$\left( {{1 \over 4}} \righ...

The solution of the differential equation
\(x{{dy} \over {dx}} + 2y\) = x2 (x \(\ne\) 0) with y(1) = 1, is :

If \(\cos x{{dy} \over {dx}} - y\sin x = 6x\), (0 < x < \({\pi \over 2}\))
and \(y\left( {{\pi \over 3}} \right)\) = 0 then \(y\left( {{\pi \over 6}} \right)\) is equal to :-

Let y = y(x) be the solution of the differential equation, \({({x^2} + 1)^2}{{dy} \over {dx}} + 2x({x^2} + 1)y = 1\) such
that y(0) = 0. If \(\sqrt ay(1)\) = \(\pi \over 32\) , then the value of
'a' is :

Let y = y(x) be the solution of the differential equation, x\({{dy} \over {dx}}\) + y = x loge x, (x > 1). If 2y(2) = loge 4 \(-\) 1, then y(e) is equal to :

If a curve passes through the point (1, –2) and has slope of the tangent at any point (x, y) on it as \({{{x^2} - 2y} \over x}\), then the curve also passes through the point :

Consider the differential equation, \({y^2}dx + \left( {x - {1 \over y}} \right)dy = 0\), If value of y is 1 when x = 1, then the value of x
for which y = 2, is :

The general solution of the differential equation (y2
– x3)dx – xydy = 0 (x \(\ne\) 0) is :
(where c is a constant of integration)

If y(x) is the solution of the differential equation \({{dy} \over {dx}} + \left( {{{2x + 1} \over x}} \right)y = {e^{ - 2x}},\,\,x > 0,\,\) where \(y\left( 1 \right) = {1 \over 2}{e^{ - 2}},\) then

The solution of the differential equation,
\({{dy} \over {dx}}\) = (x – y)2, when y(1) = 1, is :

If  \({{dy} \over {dx}} + {3 \over {{{\cos }^2}x}}y = {1 \over {{{\cos }^2}x}},\,\,x \in \left( {{{ - \pi } \over 3},{\pi \over 3}} \right)\)  and  \(y\left( {{\pi \over 4}} \right) = {4 \over 3},\)  then  $$y\left( { - {\pi \over 4}} \r...

Let f be a differentiable function such that f '(x) = 7 - \({3 \over 4}{{f\left( x \right)} \over x},\) (x > 0) and f(1) \(\ne\) 4. Then \(\mathop {\lim }\limits_{x \to 0'} \,\) xf\(\left( {{1 \over x}} \right)\) :

The curve amongst the family of curves represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1, 1) is :

If y = y(x) is the solution of the differential equation
\({{dy} \over {dx}} = \left( {\tan x - y} \right){\sec ^2}x\), \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)\),
such that y (0) = 0, then $$y\left( { - {\pi \over 4}} \r...

Let y = y(x) be the solution of the differential equation,
\({{dy} \over {dx}} + y\tan x = 2x + {x^2}\tan x\), \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)\), such that
y(0) = 1. Then :

The differential equation representing the family of ellipse having foci eith on the x-axis or on the \(y\)-axis, center at the origin and passing through the point (0, 3) is :

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} + 2y = f\left( x \right),\) where \(f\left( x \right) = \left\{ {\matrix{ {1,} & {x \in \left[ {0,1} \right]} \cr {0,} & {otherwise} \cr } } \right.\)...

The curve satifying the differeial equation, (x2 \(-\) y2) dx + 2xydy = 0 and passing through the point (1, 1) is :

Let y = y(x) be the solution of the differential equation
\(\sin x{{dy} \over {dx}} + y\cos x = 4x\), \(x \in \left( {0,\pi } \right)\).
If \(y\left( {{\pi \over 2}} \right) = 0\), then \(y\left( {{\pi \over 6}} \right)\) is equal to :

If 2x = y\({^{{1 \over 5}}}\) + y\({^{ - {1 \over 5}}}\) and
(x2 \(-\) 1) \({{{d^2}y} \over {d{x^2}}}\) + \(\lambda\)x \({{dy} \over {dx}}\) + ky = 0,
then \(\lambda\) + k is equal to :

The curve satisfying the differential equation, ydx \(-\)(x + 3y2)dy = 0 and passing through the point (1, 1), also passes through the point :

If \(\left( {2 + \sin x} \right){{dy} \over {dx}} + \left( {y + 1} \right)\cos x = 0\) and y(0) = 1,
then \(y\left( {{\pi \over 2}} \right)\) is equal to :

If   f(x) is a differentiable function in the interval (0, \(\infty\)) such that f (1) = 1 and
\(\mathop {\lim }\limits_{t \to x}\)   \({{{t^2}f\left( x \right) - {x^2}f\left( t \right)} \over {t - x}} = 1,\) for each x > 0, then $$f\l...

The solution of the differential equation
\({{dy} \over {dx}}\, + \,{y \over 2}\,\sec x = {{\tan x} \over {2y}},\,\,\)
where 0 \(\le\) x < \({\pi \over 2}\), and y (0) = 1, is given by :

If a curve \(y=f(x)\) passes through the point \((1,-1)\) and satisfies the differential equation, \(y(1+xy) dx=x\) \(dy\), then \(f\left( { - {1 \over 2}} \right)\) is equal to :

Let \(y(x)\) be the solution of the differential equation
\(\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log x,\left( {x \ge 1} \right).\) Then \(y(e)\) is equal to :

Let the population of rabbits surviving at time \(t\) be governed by the differential equation \({{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 200.\) If \(p(0)=100,\) then \(p(t)\) equals:

At present, a firm is manufacturing \(2000\) items. It is estimated that the rate of change of production P w.r.t. additional number of workers \(x\) is given by \({{dp} \over {dx}} = 100 - 12\sqrt x .\) If the firm employs \(25\) more wo...

The population \(p\) \((t)\) at time \(t\) of a certain mouse species satisfies the differential equation \({{dp\left( t \right)} \over {dt}} = 0.5\,p\left( t \right) - 450.\,\,\) If \(p(0)=850,\) then the time at which the population becom...

If \({{dy} \over {dx}} = y + 3 > 0\,\,\) and \(y(0)=2,\) then \(y\left( {\ln 2} \right)\) is equal to :

Let \(I\) be the purchase value of an equipment and \(V(t)\) be the value after it has been used for \(t\) years. The value \(V(t)\) depreciates at a rate given by differential equation $${{dv\left( t \right)} \over {dt}} = - k\left( {T - ...

Solution of the differential equation
\(\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 < x <{\pi \over 2}\) is :

The differential equation which represents the family of curves \(y = {c_1}{e^{{c_2}x}},\) where \({c_1}\) , and \({c_2}\) are arbitrary constants, is

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

The differential equation of all circles passing through the origin and having their centres on the \(x\)-axis is :

The differential equation whose solution is \(A{x^2} + B{y^2} = 1\)
where \(A\) and \(B\) are arbitrary constants is of

If \(x{{dy} \over {dx}} = y\left( {\log y - \log x + 1} \right),\) then the solution of the equation is :

The differential equation representing the family of curves \({y^2} = 2c\left( {x + \sqrt c } \right),\) where \(c>0,\) is a parameter, is of order and degree as follows:

The differential equation for the family of circle \({x^2} + {y^2} - 2ay = 0,\) where a is an arbitrary constant is :

Solution of the differential equation \(ydx + \left( {x + {x^2}y} \right)dy = 0\) is

The degree and order of the differential equation of the family of all parabolas whose axis is \(x\)-axis, are respectively.

The solution of the differential equation
\(\left( {1 + {y^2}} \right) + \left( {x - {e^{{{\tan }^{ - 1}}y}}} \right){{dy} \over {dx}} = 0,\) is :

The order and degree of the differential equation
\(\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}\) are

The solution of the equation \(\,{{{d^2}y} \over {d{x^2}}} = {e^{ - 2x}}\)