Differential Equations

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

The differential equation of all circles passing through the origin and having their centres on the $$x$$-axis 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 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 :

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

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 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, $$\int\limits_1^2 {f(x)dx = {{62} \over 5}}$$?

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

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 :

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

Let 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 $$\left|y\left(\frac{1}{8}\right)\right|$$ is equal to

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

Let a differentiable function $$f$$ satisfy f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3. Then $$12 f(8)$$ is equal to :

Let $$f(x)$$ be a positive function such that the area bounded by $$y=f(x), y=0$$ from $$x=0$$ to $$x=a>0$$ is $$e^{-a}+4 a^2+a-1$$. Then the differential equation, whose general solution is $$y=c_1 f(x)+c_2$$, where $$c_1$$ and $$c_2$$ are arbitrary constants, is

The differential equation of the family of circles passing through the origin and having centre at the line $$y=x$$ is :

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

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

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