Differential Equations Questions (544)

Let $y = y(x)$ be the solution of the differential equation $2\cos x\,\dfrac{dy}{dx} = \sin 2x - 4y\sin x$, $x\in\left(0,\dfrac{\pi}{2}\right)$. If $y\!\left(\dfrac{\pi}{3}\right) = 0$, then $y'\!\left(\dfrac{\pi}{4}\right)+y\!\left(\dfrac{\pi}{4}\right)$ is equal to ____.

Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}=\dfrac{(\tan x)+y}{\sin x(\sec x-\sin x\tan x)}$, $x\in\left(0,\dfrac{\pi}{2}\right)$ satisfying the condition $y\left(\dfrac{\pi}{4}\right)=2$. Then $y\left(\dfrac{\pi}{3}\right)$ is

Let $y = f(x)$ be the solution of the differential equation $\dfrac{dy}{dx}+\dfrac{xy}{x^2-1} = \dfrac{x^6+4x}{\sqrt{1-x^2}}$, $-1<x<1$ such that $f(0) = 0$. If $6\displaystyle\int_{-1/2}^{1/2}f(x)\,dx = 2\pi-\alpha$ then $\alpha^2$ is equal to ____.

If $x = f(y)$ is the solution of the differential equation $(1+y^2)+\left(x-2e^{\tan^{-1}y}\right)\dfrac{dy}{dx} = 0$, $y\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ with $f(0) = 1$, then $f\!\left(\dfrac{1}{\sqrt{3}}\right)$ is equal to:

If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0$, $y(0)=3$, is $\alpha x+\beta y+3\log_e|2x+3y-\gamma|=6$, then $\alpha+2\beta+3\gamma$ is equal to

Let $x = x(y)$ be the solution of the differential equation $y^2\,dx+\left(x-\dfrac{1}{y}\right)dy = 0$. If $x(1) = 1$, then $x\!\left(\dfrac{1}{2}\right)$ is:

Let \(f : [0,1] \to \mathbb{R}\) be such that \(f(xy) = f(x)\cdot f(y)\), for all \(x, y \in [0,1]\), and \(f(0) \neq 0\). If \(y = y(x)\) satisfies the differential equation, \(\dfrac{dy}{dx} = f(x)\) with \(y(0) = 1\), then \(y\!\left(\dfrac{1}{4}\right) + y\!\left(\dfrac{3}{4}\right)\) is equal to _____.

The solution of $x^2\,dy - y^2\,dx + xy(x-y)\,dy = 0$ is $\ln\left|\dfrac{x-y}{xy}\right| = \dfrac{y^k}{2} + c$, then the value of $k$ 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

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

The order of differential equation whose general solution is given by \(y = (c_1 + c_2)\sin(x + c_3) - c_4 e^{x+c_5}\) is _____.

The degree and order of the differential equation on the family of all parabolas whose axis is x-axis are, respectively,

If the general solution of the differential equation \(y' = \dfrac{y}{x} + \Phi\!\left(\dfrac{x}{y}\right)\), for some function \(\Phi\), is given by \(y\ln|cx| = x\), where \(c\) is an arbitrary constant, then \(\Phi(2)\) is equal to

The solution of $y = 2x\left(\frac{dy}{dx}\right) + x^2\left(\frac{dy}{dx}\right)^4$ 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:

The solution of $(y(1 + x^{-1}) + \sin y)dx + (x + \log_e x + x\cos y)dy = 0$ is:

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

Let \(y=f(x)\) be a differentiable function satisfying \(f(x)+f'(x)=xe^{-x}\) for all values of real \(x\). If \(f(0)=0\), then the value of \(f(1)\) equals:

The differential equation of the family of curves \(y^2 = 2c(x + \sqrt{c})\), where \(c > 0\), is a parameter, is of order and degree as follows:

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:

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:

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:

Let $y = y(t)$ be a solution to the differential equation $y' + 2ry = t^2$, then $16 \lim_{t \to \infty} \frac{y}{t}$ is ______.

If the solution of the differential equation $\frac{dy}{dx} = \frac{1}{x \cos y + \sin 2y}$ is $y = ce^{\sin y} - k(1 + \sin y)$, then the value of $k$ is ______.

The curve passing through the point $(1, 1)$ satisfies the differential equation $\frac{dy}{dx} + \frac{\sqrt{(x^2-1)(y^2-1)}}{xy} = 0$. If the curves passes through the point $\left(\sqrt{2}, k\right)$, then the value of $[k]$ is (where $[.]$ represents greatest integer function).

Find the constant of integration by the general solution of the differential equation $(2x^2y - 2y^4)dx + (x^3 + 3x^3y)dy = 0$ if curve passes through $(1, 1)$.

If $f : R - \{-1\} \to R$ and $f$ is differentiable function which satisfies: $f(x + f(y)) + xf(y) = y + f(x) + yf(x)\forall x, y \in R - \{-1\}, f(1) \neq 1$ then find the value of $2019\left[1 + f(2018)\right]$.

Let $f(x)$ be a twice differentiable bounded function satisfy $2f'(x), f''(x) + 2(f'(x))^3, f''(x) = -f''(x)$. If $f(x)$ is bounded in between $y = k_1$ and $y = k_2$. Then the number of integers between $k_i$ and $k_2$ is/are (where $f(0) = f'(0) = 0$)

Let $y=f(x)$ satisfy $\dfrac{dy}{dx}=2xe^{-y}$, $\forall x\in\mathbb{R}$. If $y'(1)=1$, then the number of solutions of $f(x)=f'(x)$ in $(0,\infty)$ is

The order of differential equation of family of circles in a plane is $m$ and highest power of second differential $\left(\frac{d^2y}{dx^2}\right)$ is $n$ then $(m+n)$ ____.

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

A function $y = f(x)$ satisfies $y'(x) = 2f(x) = x^4f^2(x), \forall x > 0$ and $f(1) = -6$. Find the value of $f\left(3^{1/3}\right)$.

The solution curve of $\dfrac{dy}{dx}=\dfrac{y^2-1}{x^2-1}$, $xy\ne 1$, passing through origin 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

The solution of $(1+y+x^2y)dx+(x+x^3)dy=0$ 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 of all non-vertical lines in a plane is

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

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 differential equation formed by eliminating constants $A$ and $B$ from $y=A\cos(\ln x)+B\sin(\ln x)$ is

Let \(y = y(x)\) be the solution of the differential equation \(\frac{dy}{dx} + 2y = f(x)\), where \[f(x) = \begin{cases} 1, & x \in [0,1] \\ 0, & \text{otherwise} \end{cases}\] If \(y(0) = 0\), then \(y\!\left(\dfrac{3}{2}\right)\) is

The general equation of an ellipse passes through the point (0, 3). The differential equation of the family of such ellipses is:

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

The population \(p(t)\) at time t of a certain mouse species satisfies the differential equation \(\dfrac{dp(t)}{dt} = 0.5\,p(t) - 450\). If \(p(0) = 850\), then the time at which the population becomes zero is

The solution of $\dfrac{dy}{dx}=\dfrac{y}{x}+\sin\dfrac{y}{x}$ is

The curve satisfying $2xy(y^2\cos(x^2y)-1) + x^2y'(y^2\cos(x^2y)+1)=0$ and passing through $(0,1)$ is

The solution of $x^2\,dy - y^2\,dx + xy(x-y)\,dy = 0$ is $\ln\left|\dfrac{x-y}{xy}\right| = \dfrac{y^k}{2} + c$, then the value of $k$ is

Let a curve \(y = f(x)\) pass through (1, 1). A tangent at point \(P(x_1, y_1)\) on the curve meets the x-axis at point \(A\) and y-axis at point \(B\). If point \(P\) divides \(AB\) in the ratio \(1:3\) and \(y(1)=1\), then \(y\left(\frac{1}{2}\right)\) equals:

The solution of the differential equation $(1-xy-x^5y^5)dx-x^2(x^4y^4+1)dy=0$ given by ($c$ is arbitrary constant)