Differential Equations

If \({y^{1/4}} + {y^{ - 1/4}} = 2x\), and \(({x^2} - 1){{{d^2}y} \over {d{x^2}}} + \alpha x{{dy} \over {dx}} + \beta y = 0\), then | \(\alpha\) \(-\) \(\beta\) | is equal to __________.

Let us consider a curve, y = f(x) passing through the point (\(-\)2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf'(x) = x2. Then :

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x\) such that y(0) = 7. Then y(\(\pi\)) is equal to :

If the solution curve of the differential equation (2x \(-\) 10y3)dy + ydx = 0, passes through the points (0, 1) and (2, \(\beta\)), then \(\beta\) is a root of the equation :

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point (2, \(-\)3) from the line 3x + 4y = 5, is given by :

If y = y(x) is the solution of the equation \({e^{\sin y}}\cos y{{dy} \over {dx}} + {e^{\sin y}}\cos x = \cos x\), y(0) = 0; then $$1 + y\left( {{\pi \over 6}} \right) + {{\sqrt 3 } \over 2}y\left( {{\pi \over 3}} \right) + {1 \over {\sqr...

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000...

The difference between degree and order of a differential equation that represents the family of curves given by \({y^2} = a\left( {x + {{\sqrt a } \over 2}} \right)\), a > 0 is _________.

Let y = y(x) be a solution curve of the differential equation \((y + 1){\tan ^2}x\,dx + \tan x\,dy + y\,dx = 0\), \(x \in \left( {0,{\pi \over 2}} \right)\). If \(\mathop {\lim }\limits_{x \to 0 + } xy(x) = 1\), then the value of $$y\left(...

Let y(x) be the solution of the differential equation 2x2 dy + (ey \(-\) 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :

Let y = y(x) be solution of the following differential equation \({e^y}{{dy} \over {dx}} - 2{e^y}\sin x + \sin x{\cos ^2}x = 0,y\left( {{\pi \over 2}} \right) = 0\) If \(y(0) = {\log _e}(\alpha + \beta {e^{ - 2}})\), then $$4(\alpha + \b...

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x < \sqrt 2 ,y(0) = 0\)then, the minimum value of \(y(x),x \in \left( { - \sqrt 2 ,\sqrt 2 } \right)\) is equal to :

Let a curve y = f(x) pass through the point (2, (loge2)2) and have slope \({{2y} \over {x{{\log }_e}x}}\) for all positive real value of x. Then the value of f(e) is equal to ______________.

Let y = y(x) be the solution of the differential equation xdy = (y + x3 cosx)dx with y(\(\pi\)) = 0, then \(y\left( {{\pi \over 2}} \right)\) is equal to :

If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is \({{{x^2} - 4x + y + 8} \over {x - 2}}\), then this curve also passes through the point :

If the curve, y = y(x) represented by the solution of the differential equation (2xy2 \(-\) y)dx + xdy = 0, passes through the intersection of the lines, 2x \(-\) 3y = 1 and 3x + 2y = 8, then |y(1)| is equal to _________.

The population P = P(t) at time 't' of a certain species follows the differential equation
\({{dP} \over {dt}}\)
= 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is :

If a curve y = f(x) passes through the point (1, 2) and satisfies \(x {{dy} \over {dx}} + y = b{x^4}\), then for what value of b, \(\int\limits_1^2 {f(x)dx = {{62} \over 5}}\)?

Let y = y(x) be the solution of the differential equation \(\left( {(x + 2){e^{\left( {{{y + 1} \over {x + 2}}} \right)}} + (y + 1)} \right)dx = (x + 2)dy\), y(1) = 1. If the domain of y = y(x) is an open interval (\(\alpha\), \(\beta\)), t...

Let y = y(x) be the solution of the differential equation \(\cos e{c^2}xdy + 2dx = (1 + y\cos 2x)\cos e{c^2}xdx\), with \(y\left( {{\pi \over 4}} \right) = 0\). Then, the value of \({(y(0) + 1)^2}\) is equal to :

Let y = y(x) be the solution of the differential equation \({e^x}\sqrt {1 - {y^2}} dx + \left( {{y \over x}} \right)dy = 0\), y(1) = \(-\)1. Then the value of (y(3))2 is equal to :

Let y = y(x) be the solution of the differential equation \(x\tan \left( {{y \over x}} \right)dy = \left( {y\tan \left( {{y \over x}} \right) - x} \right)dx\), \(- 1 \le x \le 1\), \(y\left( {{1 \over 2}} \right) = {\pi \over 6}\). Then t...

Let a curve y = y(x) be given by the solution of the differential equation \(\cos \left( {{1 \over 2}{{\cos }^{ - 1}}({e^{ - x}})} \right)dx = \sqrt {{e^{2x}} - 1} dy\). If it intersects y-axis at y = \(-\)1, and the intersection point of t...

Let y = y(x) satisfies the equation \({{dy} \over {dx}} - |A| = 0\), for all x > 0, where \(A = \left[ {\matrix{ y & {\sin x} & 1 \cr 0 & { - 1} & 1 \cr 2 & 0 & {{1 \over x}} \cr } } \right]\). If \(y(\pi ) = \pi + 2\), th...

If y = y(x) is the solution curve of the differential equation \({x^2}dy + \left( {y - {1 \over x}} \right)dx = 0\) ; x > 0 and y(1) = 1, then \(y\left( {{1 \over 2}} \right)\) is equal to :

The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is :

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} = (y + 1)\left( {(y + 1){e^{{x^2}/2}} - x} \right)\), 0 < x < 2.1, with y(2) = 0. Then the value of \({{dy} \over {dx}}\) at x = 1 is equal to :

Let y = y(x) be the solution of the differential equation xdy \(-\) ydx = \(\sqrt {({x^2} - {y^2})} dx\), x \(\ge\) 1, with y(1) = 0. If the area bounded by the line x = 1, x = e\(\pi\), y = 0 and y = y(x) is \(\alpha\)e2\(\pi\) + $$\beta...

Which of the following is true for y(x) that satisfies the differential equation \({{dy} \over {dx}}\) = xy \(-\) 1 + x \(-\) y; y(0) = 0 :

If the curve y = y(x) is the solution of the differential equation \(2({x^2} + {x^{5/4}})dy - y(x + {x^{1/4}})dx = {2x^{9/4}}dx\), x > 0 which passes through the point \(\left( {1,1 - {4 \over 3}{{\log }_e}2} \right)\), then the value of y(...

Let y = y(x) be the solution of the differential equation \(\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx,0 \le x \le {\pi \over 2},y(0) = 0\). Then, \(y\left( {{\pi \over 3}} \right)\) is equal to :

Let the curve y = y(x) be the solution of the differential equation, \({{dy} \over {dx}}\) = 2(x + 1). If the numerical value of area bounded by the curve y = y(x) and x-axis is \({{4\sqrt 8 } \over 3}\), then the value of y(1) is equal to ...

If y = y(x) is the solution of the differential equation, \({{dy} \over {dx}} + 2y\tan x = \sin x,y\left( {{\pi \over 3}} \right) = 0\), then the maximum value of the function y(x) over R is equal to:

If y = y(x) is the solution of the differential equation \({{dy} \over {dx}}\) + (tan x) y = sin x, \(0 \le x \le {\pi \over 3}\), with y(0) = 0, then \(y\left( {{\pi \over 4}} \right)\) equal to :

Let C1 be the curve obtained by the solution of differential equation \(2xy{{dy} \over {dx}} = {y^2} - {x^2},x > 0\). Let the curve C2 be the solution of \({{2xy} \over {{x^2} - {y^2}}} = {{dy} \over {dx}}\). If both the curves pass through...

If for x \(\ge\) 0, y = y(x) is the solution of the
differential equation
(x + 1)dy = ((x + 1)2 + y – 3)dx, y(2) = 0,
then y(3) is equal to _______.

If \({{dy} \over {dx}} = {{xy} \over {{x^2} + {y^2}}}\); y(1) = 1; then a value of x
satisfying y(x) = e is :

Let y = y(x) be a solution of the differential
equation,
\(\sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0\), |x| < 1.
If \(y\left( {{1 \over 2}} \right) = {{\sqrt 3 } \over 2}\), then $$y\left( { - {1 \over {\sqrt 2 }}} \right...

The differential equation of the family of
curves, x2 = 4b(y + b), b \(\in\) R, is :

If y = y(x) is the solution of the differential equation, \({e^y}\left( {{{dy} \over {dx}} - 1} \right) = {e^x}\) such that y(0) = 0, then
y(1) is equal to:

Let y = y(x) be the solution curve of the differential equation,
\(\left( {{y^2} - x} \right){{dy} \over {dx}} = 1\), satisfying y(0) =
1. This curve intersects the x-axis at a point whose abscissa is :

The general solution of the differential equation
\(\sqrt {1 + {x^2} + {y^2} + {x^2}{y^2}}\) + xy\({{dy} \over {dx}}\) = 0 is :
(where C is a constant of integration)

If \(y = \left( {{2 \over \pi }x - 1} \right) cosec\,x\) is the solution of the
differential equation, \({{dy} \over {dx}} + p\left( x \right)y = {2 \over \pi } cosec\,x\),
\(0 < x < {\pi \over 2}\), then the function p(x) is equal to :

If y = y(x) is the solution of the differential equation \({{5 + {e^x}} \over {2 + y}}.{{dy} \over {dx}} + {e^x} = 0\) satisfying
y(0) = 1, then a value of y(loge13) is :

Let y = y(x) be the solution of the differential
equation
cosx\({{dy} \over {dx}}\) + 2ysinx = sin2x, x \(\in\) \(\left( {0,{\pi \over 2}} \right)\).
If
y\(\left( {{\pi \over 3}} \right)\) = 0, then y\(\left( {{\pi \over 4}} \right)\) ...

Let y = y(x) be the solution of the differential equation, xy'- y = x2(xcosx + sinx), x > 0. if y (\(\pi\)) = \(\pi\) then \(y''\left( {{\pi \over 2}} \right) + y\left( {{\pi \over 2}} \right)\) is equal to :

The solution of the differential equation
\({{dy} \over {dx}} - {{y + 3x} \over {{{\log }_e}\left( {y + 3x} \right)}} + 3 = 0\) is:
(where c is a constant of integration)

The solution curve of the differential equation,
(1 + e-x)(1 + y2)\({{dy} \over {dx}}\) = y2,
which passes through
the point (0, 1), is :

If x3dy + xy dx = x2dy + 2y dx; y(2) = e and
x > 1, then y(4) is equal to :

Let y = y(x) be the solution of the differential
equation,
\({{2 + \sin x} \over {y + 1}}.{{dy} \over {dx}} = - \cos x\), y > 0,y(0) = 1. If y(\(\pi\)) = a and \({{dy} \over {dx}}\) at x =
\(\pi\) is b, then the ordered pair
(a, b) is eq...