Differential Equations Questions (544)

Let \(y = (a\sin x + (b+c)\cos x)e^x + d\), where a, b, c and d are parameters represent a family of curves. The differential equation for the given family of curves is \(y'' - ay' + by = 0\). Then find \(a + b\).

Let $y=y(x)$ be the solution curve of the differential equation $\dfrac{dy}{dx}=\dfrac{y}{x}(1+xy^2(1+\log_e x))$, $x>0$, $y(1)=3$. Then $\dfrac{y^2(x)}{9}$ is equal to:

Let $y=y(t)$ be a solution of the differential equation $\dfrac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$, where $\alpha>0$, $\beta>0$ and $\gamma>0$. Then $\displaystyle\lim_{t\to\infty}y(t)$:

If the general solution of a differential equation is \((y + c)^2 = cx\), where \(c\) is an arbitrary constant, then the order of the differential equation is _____.

Let $y=f(x)$ be the solution of the differential equation $y(x+1)\,dx-x^2\,dy=0$, $y(1)=e$. Then $\displaystyle\lim_{x\to0^+}f(x)$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $x\log_e x\,\dfrac{dy}{dx}+y=x^2\log_e x$, $(x>1)$. If $y(2)=2$, then $y(e)$ is equal to:

97. The general solution of the differential equation \((1 + \tan y)(dx - dy) + 2x\, dy = 0\) is:[Note: Where \(C\) is constant of integration.]

If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_e x)\,dx+x\,dy=0$, $x>0$ passes through the point $(1,1)$ and $y(e)=\dfrac{\alpha-\tan\left(\frac{3}{2}\right)}{\beta+\tan\left(\frac{3}{2}\right)}$, then $\alpha+2\beta$ is

If a curve \(y = f(x)\) passes through the point \((1, -1)\) and satisfies the differential equation, \(y(1 + xy)\,dx = x\,dy\), then \(f\!\left(-\dfrac{1}{2}\right)\) is equal to

Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10\displaystyle\int_0^x(t+2)f(t)\,dt$, $x\geq 0$. Then $f(2)$ is equal to ____.

Let $y = y(x)$ be the solution of the differential equation $\left(xy-5x^2\sqrt{1+x^2}\right)dx+\left(1+x^2\right)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to

Let the solution $y=y(x)$ of the differential equation $\dfrac{dy}{dx}-y=1+4\sin x$ satisfy $y(\pi)=1$. Then $y\left(\dfrac{\pi}{2}\right)+10$ is equal to ________.

Let $y=y(x)$ be the solution of the differential equation $(1+x^2)\dfrac{dy}{dx}+y=e^{\tan^{-1}x}$, $y(1)=0$. Then $y(0)$ is:

A tangent to the curve \(y = f(x)\) cuts the line \(y = x\) at a point which is at a distance of 1 unit from Y-axis. The equation of the curve is

If $x=x(t)$ is the solution of the differential equation $(t+1)dx=(2x+(t+1)^4)\,dt$, $x(0)=2$, then $x(1)$ equals

Let x = x(y) be the solution of the differential equation y 2 dx + (x - 1 ) dy = 0 . If x(1) = 1, then x ( 1 ) is : y 2

Let y = y(x) be the solution of the differential equation (xy - 5x \sqrt1 + x ) dx + (1 + x ) dy = 0, y(0) = 0. 2 2 2 Then y(\sqrt3) is equal to

Let $y=y(x)$ be the solution curve of the differential equation $\left(1+x^2\right)dy+\left(y-\tan^{-1}x\right)dx=0$, $y(0)=1$. Then the value of $y(1)$ is:

If x = f (y) is the solution of the differential equation -1 dy 2 tan y \pi \pi (1 + y ) + (x - 2e ) = 0, y \in (- , ) dx 2 2 with f (0) = 1, then f ( 1 ) is equal to : \sqrt3

Let $y=y(x)$ be the solution of the differential equation $(2x\log_e x)\dfrac{dy}{dx}+2y=\dfrac{3}{x}\log_e x$, $x>0$ and $y(e^{-1})=0$. Then $y(e)$ is equal to:

A function $y = f(x)$ satisfying the differential equation $\frac{dy}{dx} - \sin x - y\cos x + \sin^2 x \cdot \frac{\sin^2 x}{x^2} = 0$ is such that, $y\left(\frac{\pi}{2}\right) = \frac{2}{\pi}$, then the statement which is correct?

Let $f:(0,\infty)\to\mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2xf(x)+3$, with $f(1) = 4$. Then $2f(2)$ is equal to:

Suppose the solution of the differential equation $\dfrac{dy}{dx}=\dfrac{(2+\alpha)x-\beta y+2}{\beta x-2\alpha y-(\beta\gamma-4\alpha)}$ represents a circle passing through origin. Then the radius of this circle is:

If the solution $y=y(x)$ of the differential equation $(x^4+2x^3+3x^2+2x+2)\,dy-(2x^2+2x+3)\,dx=0$ satisfies $y(-1)=-\dfrac{\pi}{4}$, then $y(0)$ is equal to:

If $y=y(x)$ satisfies the differential equation $16\!\left(\sqrt{x+9\sqrt{x}}\right)\!\left(4+\sqrt{9+\sqrt{x}}\right)\cos y\,dy=\left(1+2\sin y\right)dx$, $x>0$ and $y(256)=\dfrac{\pi}{2}$, $y(49)=\alpha$, then $2\sin\alpha$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\sec x\,dy+\{2(1-x)\tan x+x(2-x)\}\,dx=0$ such that $y(0)=2$. Then $y(2)$ is equal to:

Let y = y(x) be the solution of the differential equation 2 cos x dy = sin 2x - 4y sin x, x \in (0, \pi ) . If dx 2 y( \pi 3 ) = 0 , then y ( ′ \pi 4 ) + y( \pi 4 ) is equal to ________ ________.

If for the solution curve y = f (x) of the differential equation dy , 2+sec x + (tan x)y = dx 2 (1+2 sec x) -\pi \sqrt3 x \in ( 2 , \pi 2 ),f ( \pi 3 ) = 10 , then f ( \pi 4 ) is equal to : \sqrt3+1

Let a curve y = f (x) pass through the points (0, 5) and (log 2, k). If the curve satisfies the differential equation e 2(3 + y)e 2x dx - (7 + e 2x ) dy = 0 , then k is equal to

98. The solution of the differential equation \(y^2\, dx + (x^2 - xy + y^2)\, dy = 0\) is:[Note: Where \(C\) is constant of integration.]

Let $\alpha$ be a non-zero real number. Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function such that $f(0)=2$ and $\lim_{x\to-\infty}f(x)=1$. If $f'(x)=\alpha f(x)+3$, for all $x\in\mathbb{R}$, then $f(-\log_e 2)$ is equal to

Let $x = x(y)$ be the solution of the differential equation $y = \left(x-y\dfrac{dx}{dy}\right)\sin\!\left(\dfrac{x}{y}\right)$, $y>0$ and $x(1) = \dfrac{\pi}{2}$. Then $\cos(x(2))$ is equal to:

If $y = y(x)$ is the solution of the differential equation $\sqrt{4-x^2}\,\dfrac{dy}{dx} = \left(\!\left(\sin^{-1}\frac{x}{2}\right)^2-y\right)\sin^{-1}\frac{x}{2}$, $-2\leq x\leq 2$, $y(2) = \dfrac{\pi^2}{4}-8$, then $y^2(0)$ is equal to ____.

Let $y=y(x)$ be the solution of the differential equation $\sec^2x\,dx+(e^{2y}\tan^2x+\tan x)\,dy=0$, $0<x<\dfrac{\pi}{2}$, $y\left(\dfrac{\pi}{4}\right)=0$. If $y\left(\dfrac{\pi}{6}\right)=\alpha$, then $e^{8\alpha}$ is equal to

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

Let $y=y(x)$ be the solution of the differential equation $(x^2+4)^2\,dy+(2x^3y+8xy-2)\,dx=0$. If $y(0)=0$, then $y(2)$ is equal to:

If for the solution curve $y = f(x)$ of the differential equation $\dfrac{dy}{dx}+(\tan x)y = \dfrac{2+\sec x}{(1+2\sec x)^2}$, $x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, $f\!\left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{10}$, then $f\!\left(\dfrac{\pi}{4}\right)$ is equal to:

The equation of a curve passing through (1, 0) for which the product of the abscissa of a point \(P\) and the intercept made by a normal at \(P\) on the \(x\)-axis equals twice the square of the radius vector of the point \(P\), is

Let $y=y(x)$ be the solution of $(x+y+2)^2dx=dy$, $y(0)=-2$. Let the maximum and minimum values of $y=y(x)$ in $\left[0,\dfrac{\pi}{3}\right]$ be $\alpha$ and $\beta$ respectively. If $(3\alpha+\pi)^2+\beta^2=\gamma+\delta\sqrt{3}$, $\gamma,\delta\in\mathbb{Z}$, then $\gamma+\delta$ equals ________.

If $\sin\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{2}$ is the solution of the differential equation $x\cos\left(\dfrac{y}{x}\right)\dfrac{dy}{dx}=y\cos\left(\dfrac{y}{x}\right)+x$ and $y(1)=\dfrac{\pi}{3}$, then $\alpha^2$ is equal to

If y = y(x) is the solution of the differential equation, 2 2 dy , then y (0) is equal to -1 x -1 x \pi -8 2 \sqrt4 - x 2 = ((sin ( )) - y) sin ( ), -2 \le x \le 2, y(2) = dx 2 2 4 6

Let a curve $y = f(x)$ pass through the points $(0,5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3+y)e^{2x}\,dx-(7+e^{2x})\,dy = 0$, then $k$ is equal to

Let $y=y_1(x)$ and $y=y_2(x)$ be solution curves of $\dfrac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$ and $y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and $y=y_2(x)$ intersect at

If the solution curve of the differential equation $\dfrac{dy}{dx}=\dfrac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan^{-1}\left(\dfrac{y-1}{x-1}\right)-\dfrac{1}{\beta}\log_e\left(\alpha+\left(\dfrac{y-1}{x-1}\right)^2\right)=\log_e|x-1|$, then $5\beta+\alpha$ is equal to

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: