Differential Equations Questions (544)

The general solution of the differential equation $x\left(\frac{dy}{dx}\right) = y\log\left(\frac{y}{x}\right)$ is:

If $f(x) = x + \int_1^x\frac{f(t)}{t}dt$, then $\int_0^{\pi}\frac{(f(\sin\theta) - \sin\theta)}{\sin\theta}d\theta$ is equal to:

If $\frac{dy}{dx} + y\frac{dx}{dy} = x.y(-2) = 1$, then :

If $(2xy-y^2-y)dx=(2xy+x-x^2)dy$ and $y(1)=1$, then the value of $12|y(-1)|$ is

The singular solution of the differential equation given in previous problem is :

Non-singular solution of the differential equation $x\frac{dy}{dx} = x - \left(\frac{dy}{dx}\right)^2$ is :

The solution of $\frac{xdx + ydy}{xdy - ydx} = \frac{a^2 - x^2 - y^2}{x^2 + y^2}$ is:

The curve $y = f(x)$ is such that the area of the trapezium formed by the coordinate axes ordinate of an arbitrary point and the tangent at this point equals half the square of its abscissa. The curve is:

The real value of $m$ for which the substitution $y = u^m$ will transform the differential equation $2x^3y\frac{dy}{dx} + y^4 = 4x^6$ into a homogeneous equation is:

Solution of the differential equation $x = 1 + xy\frac{dy}{dx} + \frac{x^2y^2}{2!}\left(\frac{dy}{dx}\right)^2 + \frac{x^3y^3}{3!}\left(\frac{dy}{dx}\right)^3 + ......$ is:

The solution of the differential equation $\frac{dy}{dx} = -\frac{1}{xy(x^2 \sin y^2 + 1)}$ is :

The general equation of the equation $y = px + \log p$ which does not contain the singular solution, is :

Solution of the differential equation $x\cos\left(\frac{x}{y}\right)(ydx + xdy) = y\sin\left(\frac{x}{y}\right)(xdy - ydx)$ is :

Solution of the differential equation $\left\{1 - \frac{y^2}{x(x-y)^2}\right\}dx + \left\{\frac{x^2}{(x-y)^2} - \frac{1}{y}\right\}dy = 0$ is :

If the solution of the differential equation $\frac{xdx - ydy}{xdy - ydx} = \sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$ be $f(x,y) + \sqrt{1+f(x,y)} = c\left(\sqrt{\frac{x+y}{\sqrt{f(x,y)}}}\right)$, then $f(x,y)$ is:

The solution of the equation $\int_0^x y(t)dt = (x+1)\int_0^x ty(t)dt, x > 0$ as $y = f(x)$ is:

Solution of the equation $\frac{xdx + ydy}{xdy - ydx} = \sqrt{\frac{a^2-x^2-y^2}{x^2+y^2}}$ is:

Through any point $(x, y)$ of a curve which passes through the origin, lines are drawn parallel to the co-ordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of:

The orthogonal trajectories of the family of coaxial circles $x^2 + y^2 + 2gx + C = 0$, where $g$ is a parameter are

A curve $f(x)$ passes through the point $P(1,1)$. The normal to the curve at point $P$ is $a(y-1) + (x-1) = 0$. If the slope of the tangent at any point on the curve is proportional to the ordinate at that point, then the equation of the curve is

A tangent drawn to the curve $y = f(x)$ at P(x, y) cuts the x-axis and y-axis at A and B respectively such that BP: AP = 3:1, given that $f(1) = 1$, then:

If $f(x), g(x)$ be twice differentiable functions on [0, 2] satisfying $f''(x) = g''(x)$, $f'(1) = 2g'(1) = 4$ and $f(2) = 3g(2) = 9$, then:

Given the differential equation $\frac{dy}{dx} = \frac{6x^2}{2y + \cos y}$, $y(1) = \pi$ and the following statements

The equation of the curve satisfying the differential equation $y_2(x^2 + 1) = 2xy_1$ passing through the point (0, 1) and having slope of tangent at $x = 0$ as 3(where $y_2$ and $y_1$ represents $2^{nd}$ and $1^{st}$ order derivative), then:

For the central conics having their axes along the coordinates:

For the differential equation $(3x + 2y^2)dx + 2x(2x + 3y^2)dy = 0$

If $y_1, y_2$ are the solution of the differential equation $\frac{dy}{dx} + P(x)y = Q(x)$, then:

If the rate at which a substance cools in moving air is proportional to the difference between the temperature of the substance and that of the air. If the temperature of air is 30°C and the substance cools from 37°C to 34°C in 15 min then:

A right circular cylinder with radius $R$ and height $H$ contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant = $K > 0$). If $T$ is time after which cylinder will be empty, then

Let $S_1 = x^2 + y^2 - kx = 0$ and $S_2 = x^2 - y^2 - cx = 0$, then

If a curve $y = f(x)$, passing through the point $(2, 1)$ satisfies the condition that length of subtangent is equal to slope of tangent in $1^{st}$ quadrant given that $\frac{dy}{dx} > 0$, then:

Consider the differential equation $\cos^2 x \frac{dy}{dx} - (\tan 2x)y = \cos^4 x$; $|x| < \frac{\pi}{4}$ and $y\left(\frac{\pi}{6}\right) = \frac{3\sqrt{3}}{8}$ then:

If the length of perpendicular from origin to any normal to the curve $y = f(x)$ is equal to its $y$ intercept, then

If $f(x) = \int_1^x \frac{\log t}{1 + t + t^2} dt, x \geq 1$, then:

If $f(x)$ is a function such that $\int_0^x (1-t)f(t)dt = \int_0^x f(t)dt; f(1) = 1$, then:

The solutions of $y = x\left(\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^3\right)$ are given by (where $p = \frac{dy}{dx}$ and $k$ is constant)

If $|y| = f(x)$ is solution of $\frac{d^2y}{dx^2} = \frac{x^3}{y^3}\frac{d^2x}{dy^2}$ such that $f(0) = 2$ and $y = g(x)$ is solution of $\frac{d^2y}{dx^2} + \frac{8y^3}{x^3} + \frac{d^2x}{dy^2} = 0$ such that $g(1) = \frac{1}{3}$, then:

If $y = e^{-x}\sin x$ and $y_n + a_ny_{n-1} = 0$ where $a_n$ is constant for $n \in \mathbb{N}$ & $y_n = \frac{d^ny}{dx^n}$ (nth derivative of $y$), then:

If $y = f(x); f'(x) \geq 0$ & $f(0) = 0$ bounding a curvilinear trapezoid with base $[0, x]$ whose area is proportional to $3^{rd}$ power of $f(x)$. If $f(1) = 3$, then:

If differential equation of the curve $y = ae^{3x} + be^{2x} + ce^x$ is $\frac{d^3y}{dx^3} + m\frac{d^2y}{dx^2} + n\frac{dy}{dx} + p y = 0$, then:

If right circular cone with radius 18 & height 27 contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant = $k > 0$). If volume of liquid is $V$ & $r$ is radius of surface of liquid left, then:

If a tangent drawn to the curve $y = f(x)$ at $(x, y)$ cuts the $x-$ axis and $y-$ axis at $A$ and $B$ respectively such that $\frac{BP}{AP} = \frac{3}{1}$ given $f(1) = 1$, then:

If $\frac{dy}{dx} = \frac{x^2-y}{x+y}$ such that $y = f(x)$ is a solution of differential equation & $f(0) = 0$, then:

Let $C$ be a curve such that the normal at any point $P$ on it meets $x-$axis $y-$axis at $A$ and $Y$ respectively. If $BP : PA = 1:2$ (internally) and the curve passes through the point $(0,4)$ then which of the following alternative(s) is/are correct?

A differentiable function satisfies $f(x) = \int_0^x [f(t)\cos t - \cos(t-x)]dt$. which is of the following hold good?

Let $\frac{dy}{dx}+y = f(x)$ where $y$ is a continuous function of $x$ with $y(0) = 1$ and $f(x) = \begin{cases} e^{-x}, & \text{if } x \le 2 \\ e^{-2}, & \text{if } x > 2 \end{cases}$. Which is of the following hold(s) good?

The general solution of the differential equation $x\left(\frac{dy}{dx}\right) = y\log\left(\frac{y}{x}\right)$ is:

Let $y = f(x)$ be a curve in the first quadrant such that the triangle formed by the co-ordinate axis and the tangent at any point on the curve has area 2. If $f(1) = 1$, then $y(2) = $

If $f(x) = x + \int_1^x\frac{f(t)}{t}dt$, then $\int_0^{\pi}\frac{(f(\sin\theta) - \sin\theta)}{\sin\theta}d\theta$ is equal to:

If $\frac{dy}{dx} + y\frac{dx}{dy} = x.y(-2) = 1$, then :