Differential Equations

Let \(y=y(x)\) be a solution curve of the differential equation.
\(\left(1-x^{2} y^{2}\right) d x=y d x+x d y\).
If the line \(x=1\) intersects the curve \(y=y(x)\) at \(y=2\) and the line \(x=2\) intersects the curve \(y=y(x)\) at $$y=\alp...

Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}+\frac{5}{x\left(x^{5}+1\right)} y=\frac{\left(x^{5}+1\right)^{2}}{x^{7}}, x > 0\). If \(y(1)=2\), then \(y(2)\) is equal to :

Let \(f\) be a differentiable function such that \({x^2}f(x) - x = 4\int\limits_0^x {tf(t)dt}\), \(f(1) = {2 \over 3}\). Then \(18f(3)\) is equal to :

Let the tangent at any point P on a curve passing through the points (1, 1) and \(\left(\frac{1}{10}, 100\right)\), intersect positive \(x\)-axis and \(y\)-axis at the points A and B respectively. If \(\mathrm{PA}: \mathrm{PB}=1: k\) and $$...

Let \({{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R\), represents a circle with center (\(\alpha\), \(\beta\)). Then, \(\alpha\) + 2\(\beta\) is equal to :

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 < x < {\pi \over 2}\) with $$y\left( {{\pi \over 4}} \right)...

Let the solution curve of the differential equation
\(x{{dy} \over {dx}} - y = \sqrt {{y^2} + 16{x^2}}\), \(y(1) = 3\) be \(y = y(x)\). Then y(2) is equal to:

Let y = y(x), x > 1, be the solution of the differential equation \((x - 1){{dy} \over {dx}} + 2xy = {1 \over {x - 1}}\), with \(y(2) = {{1 + {e^4}} \over {2{e^4}}}\). If \(y(3) = {{{e^\alpha } + 1} \over {\beta {e^\alpha }}}\), then the va...

If y = y(x) is the solution of the differential equation \(\left( {1 + {e^{2x}}} \right){{dy} \over {dx}} + 2\left( {1 + {y^2}} \right){e^x} = 0\) and y (0) = 0, then $$6\left( {y'(0) + {{\left( {y\left( {{{\log }_e}\sqrt 3 } \right)} \righ...

Let the solution curve \(y=y(x)\) of the differential equation \(\left(1+\mathrm{e}^{2 x}\right)\left(\frac{\mathrm{d} y}{\mathrm{~d} x}+y\right)=1\) pass through the point \(\left(0, \frac{\pi}{2}\right)\). Then, $$\lim\limits_{x \rightarr...

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}+\left(\frac{2 x^{2}+11 x+13}{x^{3}+6 x^{2}+11 x+6}\right) y=\frac{(x+3)}{x+1}, x>-1\), which passes through the point \((0,1)\). Then \(y(1)\) is equal to ...

If the solution curve of the differential equation \(\frac{d y}{d x}=\frac{x+y-2}{x-y}\) passes through the points \((2,1)\) and \((\mathrm{k}+1,2), \mathrm{k}>0\), then

Let y = y(x) be the solution of the differential equation \(x(1 - {x^2}){{dy} \over {dx}} + (3{x^2}y - y - 4{x^3}) = 0\), \(x > 1\), with \(y(2) = - 2\). Then y(3) is equal to :

Let the solution curve \(y = y(x)\) of the differential equation
\(\left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]x{{dy} \over {dx}} = x + \left[ {{x \over {\sqrt {{x^2} - {y^2}} }} + {e^{{y \over x}}}} \right]y\)
pa...

Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 \(\tan x(\cos x - y)\). If the curve passes through the point \(\left( {{\pi \over 4},0} \right)\), then the value of \(\int\limits_0^{\pi /2} {y\,dx}\) is equal to ...

Let x = x(y) be the solution of the differential equation \(2y\,{e^{x/{y^2}}}dx + \left( {{y^2} - 4x{e^{x/{y^2}}}} \right)dy = 0\) such that x(1) = 0. Then, x(e) is equal to :

If \(y=y(x), x \in(0, \pi / 2)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2 \mathrm{e}^{-4 x}(2 \sin 2 x+\cos 2 x)\), with $$y(\pi / 4)=\mathrm...

Let the solution curve of the differential equation \(x \mathrm{~d} y=\left(\sqrt{x^{2}+y^{2}}+y\right) \mathrm{d} x, x>0\), intersect the line \(x=1\) at \(y=0\) and the line \(x=2\) at \(y=\alpha\). Then the value of \(\alpha\) is :

The differential equation of the family of circles passing through the points \((0,2)\) and \((0,-2)\) is :

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}+\frac{1}{x^{2}-1} y=\left(\frac{x-1}{x+1}\right)^{1 / 2}\), \(x >1\) passing through the point \(\left(2, \sqrt{\frac{1}{3}}\right)\). Then $$\sqrt{7}\, y...

If \({{dy} \over {dx}} + {{{2^{x - y}}({2^y} - 1)} \over {{2^x} - 1}} = 0\), x, y > 0, y(1) = 1, then y(2) is equal to :

Let \({{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}\), where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is :

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

If the solution curve of the differential equation \((({\tan ^{ - 1}}y) - x)dy = (1 + {y^2})dx\) passes through the point (1, 0), then the abscissa of the point on the curve whose ordinate is tan(1), is

Let \(y=y(x)\) be the solution curve of the differential equation
\(\sin \left( {2{x^2}} \right){\log _e}\left( {\tan {x^2}} \right)dy + \left( {4xy - 4\sqrt 2 x\sin \left( {{x^2} - {\pi \over 4}} \right)} \right)dx = 0\), $$0 < x < \sqrt ...

Let \(y=y_{1}(x)\) and \(y=y_{2}(x)\) be two distinct solutions of the differential equation \(\frac{d y}{d x}=x+y\), with \(y_{1}(0)=0\) and \(y_{2}(0)=1\) respectively. Then, the number of points of intersection of \(y=y_{1}(x)\) and $$y=...

Let \(S = (0,2\pi ) - \left\{ {{\pi \over 2},{{3\pi } \over 4},{{3\pi } \over 2},{{7\pi } \over 4}} \right\}\). Let \(y = y(x)\), x \(\in\) S, be the solution curve of the differential equation $${{dy} \over {dx}} = {1 \over {1 + \sin 2x}}...

Let the solution curve y = y(x) of the differential equation \((4 + {x^2})dy - 2x({x^2} + 3y + 4)dx = 0\) pass through the origin. Then y(2) is equal to _____________.

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

If \(y = y(x)\) is the solution of the differential equation \(x{{dy} \over {dx}} + 2y = x\,{e^x}\), \(y(1) = 0\) then the local maximum value of the function \(z(x) = {x^2}y(x) - {e^x},\,x \in R\) is :

Let a curve \(y=y(x)\) pass through the point \((3,3)\) and the area of the region under this curve, above the \(x\)-axis and between the abscissae 3 and \(x(>3)\) be \(\left(\frac{y}{x}\right)^{3}\). If this curve also passes through the p...

If \({{dy} \over {dx}} + 2y\tan x = \sin x,\,0 < x < {\pi \over 2}\) and \(y\left( {{\pi \over 3}} \right) = 0\), then the maximum value of \(y(x)\) is :

Suppose \(y=y(x)\) be the solution curve to the differential equation \(\frac{d y}{d x}-y=2-e^{-x}\) such that \(\lim\limits_{x \rightarrow \infty} y(x)\) is finite. If \(a\) and \(b\) are respectively the \(x\) - and \(y\)-intercepts of t...

Let the solution curve \(y=f(x)\) of the differential equation \(\frac{d y}{d x}+\frac{x y}{x^{2}-1}=\frac{x^{4}+2 x}{\sqrt{1-x^{2}}}\), \(x\in(-1,1)\) pass through the origin. Then $$\int\limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}...

If the solution curve \(y = y(x)\) of the differential equation \({y^2}dx + ({x^2} - xy + {y^2})dy = 0\), which passes through the point (1, 1) and intersects the line \(y = \sqrt 3 x\) at the point \((\alpha ,\sqrt 3 \alpha )\), then value...

Let \(y = y(x)\) be the solution of the differential equation \((x + 1)y' - y = {e^{3x}}{(x + 1)^2}\), with \(y(0) = {1 \over 3}\). Then, the point \(x = - {4 \over 3}\) for the curve \(y = y(x)\) is :

Let \(g:(0,\infty ) \to R\) be a differentiable function such that $$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} +...

If \(y = y(x)\) is the solution of the differential equation \(2{x^2}{{dy} \over {dx}} - 2xy + 3{y^2} = 0\) such that \(y(e) = {e \over 3}\), then y(1) is equal to :

The general solution of the differential equation \(\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0\) is :

The slope of the tangent to a curve \(C: y=y(x)\) at any point \((x, y)\) on it is \(\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 x}}\).

If \(C\) passes through the points $$\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\r...

Let \(y=y(x)\) be the solution of the differential equation
\(\frac{d y}{d x}=\frac{4 y^{3}+2 y x^{2}}{3 x y^{2}+x^{3}}, y(1)=1\).
If for some \(n \in \mathbb{N}, y(2) \in[n-1, n)\), then \(n\) is equal to _____________.

Let a smooth curve \(y=f(x)\) be such that the slope of the tangent at any point \((x, y)\) on it is directly proportional to \(\left(\frac{-y}{x}\right)\). If the curve passes through the points \((1,2)\) and \((8,1)\), then $$\left|y\left...

If x = x(y) is the solution of the differential equation \(y{{dx} \over {dy}} = 2x + {y^3}(y + 1){e^y},\,x(1) = 0\); then x(e) is equal to :

If \({{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}\), y(0) = 1, then y(1) is equal to :

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

If \({{dy} \over {dx}} = {{{2^x}y + {2^y}{{.2}^x}} \over {{2^x} + {2^{x + y}}{{\log }_e}2}}\), y(0) = 0, then for y = 1, the value of x lies in the interval :

If \(y = y(x),y \in \left[ {0,{\pi \over 2}} \right)\) is the solution of the differential equation \(\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0\), with y(0) = 0, then \(5y'\left( {{\pi \over 2}} \right)\) is equal to _____...

Let y = y(x) be solution of the differential equation \({\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y\), with y(0) = 0.If \(y\left( { - {2 \over 3}{{\log }_e}2} \right) = \alpha {\log _e}2\), then the value of \(\alpha\) is equal t...

Let y = y(x) be the solution of the differential equation dy = e\(\alpha\)x + y dx; \(\alpha\) \(\in\) N. If y(loge2) = loge2 and y(0) = loge\(\left( {{1 \over 2}} \right)\), then the value of \(\alpha\) is equal to _____________.

Let y = y(x) be the solution of the differential equation (x \(-\) x3)dy = (y + yx2 \(-\) 3x4)dx, x > 2. If y(3) = 3, then y(4) is equal to :