Differential Equations Questions (544)

If the solution curve $f(x,y)=0$ of the differential equation $(1+\ln x)\dfrac{dy}{dx}-x\ln x=e^y$, $x>0$, passes through $(1,0)$ and $(a,2)$, then $a^a$ is equal to

The solution of the differential equation $\dfrac{dy}{dx}=-\left(\dfrac{x^2+3y^2}{3x^2+y^2}\right)$, $y(1)=0$ is:

The function $f(x)$ satisfying the equation $[f'(x)]^2+4f'(x)f(x)+[f(x)]^2 = 0$

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

Let $y=y(x)$ be the solution of the differential equation $x\dfrac{dy}{dx}-\sin 2y=x^3\left(2-x^3\right)\cos^2 y$, $x\neq0$. If $y(2)=0$, then $\tan(y(1))$ is equal to

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

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) = $

The solution of $\frac{dy}{x^2 + y^2} = \left(\frac{1}{x^2 + y^2} - 1\right) dx$ is:

Which of the following pair $(s)$ is/are orthogonal?

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

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:

If the independent variable $x$ is changed to $y$, then the differential equation $x\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 - \frac{dy}{dx} = 0$ is changed to $x\frac{d^2x}{dy^2} + \left(\frac{dx}{dy}\right)^2 = k$ where $k$ equals ______.

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

Identify the statement(s) which is/are true?

Let $\alpha x=\exp(x^\beta y^\gamma)$ be the solution of the differential equation $2x^2y\,dy-(1-xy^2)\,dx=0$, $x>0$, $y(2)=\sqrt{\log_e 2}$. Then $\alpha+\beta-\gamma$ equals:

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is

Let $y=y(x)$ satisfy the differential equation $\left(2xy + x^2y + \dfrac{y^3}{3}\right)dx + \left(x^2+y^2\right)dy=0$. If $y(1)=1$ and $(y(0))^3=ke$, $k\in\mathbb{N}$, then $k$ is

\\(\\cos x\\,\\dfrac{dy}{dx}+y\\sin x=1\\), \\(y(0)=1\\). Find \\(y(\\pi/3)\\).

Ajay takes 100 mg paracetamol every 12 hours. Amount halved every 5 hours. Amount in bloodstream 15 hours after first dose is (mg)

Curve $y=f(x)$ through origin satisfies $\dfrac{7dy}{dx}+\dfrac{28x^3y}{1+x^4}=\dfrac{40x^4}{1+x^4}$. If area enclosed by $y=f^{-1}(x)$, x-axis, and $x=4/7$ in 1st quadrant is $A$, then $42A$ is

Let \(f(x)\) be a polynomial function satisfying \(f'(x) + f(x) = x\). Then the value of \(f(4)\) is equal to:

A curve passes through $\left(1,\dfrac{\pi}{6}\right)$. Let the slope at each point $(x,y)$ be $\dfrac{y}{x}+\sec\!\left(\dfrac{y}{x}\right)$, $x>0$. The equation of the curve is

The solution of the differential equation $\dfrac{dy}{dx}=(4x+y+1)^2$ is

The solution of the differential equation $x\,dy + y\,dx = 0$ passes through the point $(2, 8)$. The latus rectum of the conic represented by the solution curve equals

Ajay takes 100 mg paracetamol every 12 hours. Amount halved every 5 hours. Amount in bloodstream 15 hours after first dose is (mg)

Curve $y=f(x)$ through origin satisfies $\dfrac{7dy}{dx}+\dfrac{28x^3y}{1+x^4}=\dfrac{40x^4}{1+x^4}$. If area enclosed by $y=f^{-1}(x)$, x-axis, and $x=4/7$ in 1st quadrant is $A$, then $42A$ is

Let $y=y(x)$ be a solution of $\dfrac{dy}{dx} + \dfrac{y\cos x}{\sin x+\cos x} = \dfrac{2\tan^2 x+\tan x+2}{\sqrt{\sin x+\cos x}}$. If $y\!\left(\dfrac{\pi}{4}\right)=2^{3/4}$, then $y^2\!\left(\dfrac{7\pi}{12}\right)$ is

The given differential equation is \(\frac{dp(t)}{dt} = \frac{1}{2}p(t) - 200\). If the initial number of rabbits is 100 and is decreasing, then \(p(t)\) is:

The differential equation whose solution is \(Ax^2 + By^2 = 1\), where \(A\) and \(B\) are arbitrary constants is of

The curve satisfying the differential equation \((x^2 - y^2)\,dx + 2xy\,dy = 0\) and passing through the point \((1,\,1)\) is

Given \(y^2\,dx + \left(x - \dfrac{1}{y}\right)dy = 0\)Find the solution of the differential equation.

The differential equation of the family of curves for which the length of the normal is equal to a constant k, is given by

If \(x\dfrac{dy}{dx} = y(\log y - \log x + 1)\), then the solution of the equation is

We have \( y = c_1 e^{c_2 x} \) Which of the following is the differential equation satisfied by this?

The solution of the differential equation \(x\dfrac{dy}{dx} + 2y = x^2\;(x \neq 0)\) with \(y(1) = 1\), is:

The general solution of the differential equation, \[\sin 2x\left(\frac{dy}{dx} - \sqrt{\tan x}\right) - y = 0,\] is

Given \(y\left(\dfrac{d^2y}{dx^2}\right) = 2\left(\dfrac{dy}{dx}\right)^2\) and the curve passes through \((2, 2)\) and \(\left(8, \dfrac{1}{2}\right)\). Find \(f(10)\). (Answer: 0.40)

If \(y(x)\) is the solution of the differential equation \((x+2)\frac{dy}{dx} = x^2 + 4x - 9,\ x \neq -2\) and \(y(0) = 0\), then \(y(-4)\) is equal to

Given inequality can be written as: \(f''(x) - 2f'(x) \geq 3(f'(x) - 2f(x))\).Let \(f'(x) - 2f(x) = g(x)\). It is given that \(g(x)e^{-3x}\) is non-decreasing, \(g(0) = f'(0) - 2f(0) = -2\). If \(f(x) \geq 3e^{2x} - 2e^{3x},\ \forall x \geq 0\), and comparing \(ah(bx) - bh(ax)\) with \(3e^{2x} - 2e^{3x}\), find \((a+b)h(0)\).

If \(f(x) = f'(x)\) and \(f(1) = 2\), then \(\dfrac{[f(3)]}{2}\). Where [.] is G.I.F.

The solution of $y\,dx-x\,dy=\sqrt{x^2+y^2}\,dx$ is

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

If \(\frac{dy}{dx} = y + 3\) and \(y(0) = 2\), then \(y(\ln 2)\) is equal to:

The solution of ODE $\dfrac{dy}{dx}=\dfrac{y(2y-x)}{x(2y+x)}$ is