Mathematics
Differential Equations
Previous Years Questions

## Numerical

If $y(x)$ is the solution of the differential equation $$x d y-\left(y^{2}-4 y\right) d x=0 \text { for } x > 0, y(1)=2,$$ and the slope of the c...
Let f : R $$\to$$ R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation $${{dy} \over {dx}} = (2 + 5y)(5y - ... Let$$f:[1,\infty ) \to [2,\infty )$$be a differentiable function such that$$f(1) = 2$$. If$$6\int\limits_1^x {f(t)dt = 3xf(x) - {x^3} - 5} $$for ... Let$$y'\left( x \right) + y\left( x \right)g'\left( x \right) = g\left( x \right),g'\left( x \right),y\left( 0 \right) = 0,x \in R,$$where$$f'(x)$$... ## MCQ (More than One Correct Answer) For x \in \mathbb{R}, let the function y(x) be the solution of the differential equation$$ \frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\righ...
Let $$\Gamma$$ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersec...
If $$g(x) = \int_{\sin x}^{\sin (2x)} {{{\sin }^{ - 1}}} (t)\,dt$$, then
A solution curve of the differential equation $$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$$ $$x>0,$$ passes through ...
Let $$f:(0,\infty ) \to R$$ be a differentiable function such that $$f'(x) = 2 - {{f(x)} \over x}$$ for all $$x \in (0,\infty )$$ and $$f(1) \ne 1$$. ...
Let $$y(x)$$ be a solution of the differential equation $$\left( {1 + {e^x}} \right)y' + y{e^x} = 1.$$ If $$y(0)=2$$, then which of the following sta...
Consider the family of all circles whose centres lie on the straight line $$y=x,$$ If this family of circle is represented by the differential equatio...
If $$y(x)$$ satisfies the differential equation $$y' - y\,tan\,x = 2x\,secx$$ and $$y(0)=0,$$ then
A curve $$y=f(x)$$ passes through $$(1,1)$$ and at $$P(x,y),$$ tangent cuts the $$x$$-axis and $$y$$-axis at $$A$$ and $$B$$ respectively such that $... The differential equation representing the family of curves $${y^2} = 2c\left( {x + \sqrt c } \right),$$ where $$c$$ is a positive parameter, is of ... ## MCQ (Single Correct Answer) If y = y(x) satisfies the differential equation$${8\sqrt x \left( {\sqrt {9 + \sqrt x } } \right)dy = {{\left( {\sqrt {4 + \sqrt {9 + \sqrt x } } } \r... The function$$y=f(x)$$is the solution of the differential equation$${{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - ... A curve passes through the point $$\left( {1,{\pi \over 6}} \right)$$. Let the slope of the curve at each point $$(x,y)$$ be $${y \over x} + \sec \l... Let a solution$$y=y(x)$$of the differential equation$$x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0$$satify$$y\left( 2 \right) = {2 \ov... If $$y=y(x)$$ and it follows the relation $$x\cos \,y + y\,cos\,x = \pi$$ then $$y''(0)=$$ For the primitive integral equation $$ydx + {y^2}dy = x\,dy;$$ $$x \in R,\,\,y > 0,y = y\left( x \right),\,y\left( 1 \right) = 1,$$ then $$y(-3)$$... The solution of primitive integral equation $$\left( {{x^2} + {y^2}} \right)dy = xy$$ $$dx$$ is $$y=y(x),$$ If $$y(1)=1$$ and $$\left( {{x_0}} \right)... The differential equation$${{dy} \over {dx}} = {{\sqrt {1 - {y^2}} } \over y}$$determines a family of circles with If$$y=y(x)$$and$${{2 + \sin x} \over {y + 1}}\left( {{{dy} \over {dx}}} \right) = - \cos x,y\left( 0 \right) = 1,$$then$$y\left( {{\pi \over ... If $$y(t)$$ is a solution of $$\left( {1 + t} \right){{dy} \over {dt}} - ty = 1$$ and $$y\left( 0 \right) = - 1,$$ then $$y(1)$$ is equal to If $${x^2} + {y^2} = 1,$$ then A solution of the differential equation $${\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0$$ is The order of the differential equation whose general solution is given by $$y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{... ## Subjective Match the statements/expressions in Column$$I$$with the open intervals in Column$$II$$. Column$$I$$(A)$$\,\,\,$$Interval contained in the domai... If length of tangent at any point on the curve$$y=f(x)$$intecepted between the point and the$$x$$-axis is length$$1.$$Find the equation of the cu... A curve$$'C''$$passes through$$(2,0)$$and the slope at$$(x,y|)$$as$$\,{{{{\left( {x + 1} \right)}^2} + \left( {y - 3} \right)} \over {x + 3}}$$... A right circular cone with radius$$R$$and height$$H$$contains a liquid which eveporates at a rate proportional to its surface area in contact with... A hemispherical tank of radius$$2$$metres is initially full of water and has an outlet of$$12$$cm2 cross-sectional area at the bottom. The outlet ... Let$$u(x)$$and$$v(x)$$satisfy the differential equation$${{du} \over {dx}} + p\left( x \right)u = f\left( x \right)$$and$${{dv} \over {dx}} + p... Determine the equation of the curve passing through the origin, in the form $$y=f(x),$$ which satisfies the differential equation $${{dy} \over {dx}} ... Let$$y=f(x)$$be a curve passing through$$(1,1)$$such that the triangle formed by the coordinate axes and the tangent at any point of the curve lie... A normal is drawn at a point$$P(x,y)$$of a curve. It meets the$$x$$-axis at$$Q.$$If$$PQ$$is of constant length$$k,$$then show that the differ... If$$\left( {a + bx} \right){e^{y/x}} = x,$$then prove that$${x^3}{{{d^2}y} \over {d{x^2}}} = {\left( {x{{dy} \over {dx}} - y} \right)^2}$\$
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