Differential Equations

The curve satisfying the differential equation, $y\,dx-(x+3y^2)\,dy=0$ and passing through the point (1, 1), also passes through the point :

A tangent to the curve, $y=f(x)$ at P(x, y) meets $x$-axis at A and $y$-axis at B. If AP : BP = 1 : 3 and $f(1)=1$, then the curve also passes through the point :

Let $f:(0,\infty)\to\mathbb R$ be a differentiable function such that $f'(x)=2-\dfrac{f(x)}{x}$ for all $x\in(0,\infty)$ and $f(1)\ne1$. Then

A solution curve of the differential equation $(x^2+xy+4x+2y+4)\dfrac{dy}{dx}-y^2=0$, $x>0$, passes through the point $(1,3)$. Then the solution curve

If a curve y = f(x) passes through the point (1, −1) and satisfies the differential equation, $y(1+xy)\,dx=x\,dy$, then $f\left(-\dfrac12\right)$ is equal to :

If f(x) is a differentiable function in the interval (0, ∞) such that f(1) = 1 and $\displaystyle\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=1$, for each x > 0, then $f\left(\dfrac32\right)$ is equal to :

For x ∈ R, x ≠ 0, if y(x) is a differentiable function such that $x\displaystyle\int_1^xy(t)\,dt=(x+1)\int_1^xt\,y(t)\,dt$, then y(x) equals :

(where C is a constant.)

The solution of the differential equation $\dfrac{dy}{dx}+\dfrac y2\sec x=\dfrac{\tan x}{2y}$, where $0\le x<\dfrac\pi2$, and y(0) = 1, is given by :

Let $y(x)$ be a solution of the differential equation $(1+e^x)y'+ye^x=1$. If $y(0)=2$, then which of the following statements is (are) true?

Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $Py''+Qy'+1=0$, where $P$, $Q$ are functions of $x$, $y$ and $y'$ (here $y'=\dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$), then which of the following statements is (are) true?

Let $y(x)$ be the solution of the differential equation
$$(x\log x)\frac{dy}{dx}+y=2x\log x,\quad(x\ge1).$$
Then $y(e)$ is equal to :

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

The solution of the differential equation $y\,dx-(x+2y^2)\,dy=0$ is $x=f(y)$. If $f(-1)=1$, then $f(1)$ is equal to :

The function $y=f(x)$ is the solution of the differential equation
$$\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$$
in $(-1,1)$ satisfying $f(0)=0$. Then
$$\int_{-\frac{\sqrt3}{2}}^{\frac{\sqrt3}{2}}f(x)\,dx$$
is

Let $f:[0,2]\to\mathbb R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let
$$F(x)=\int_0^{x^2}f(\sqrt t)\,dt$$
for $x\in[0,2]$. If $F'(x)=f'(x)$ for all $x\in(0,2)$, then $F(2)$ equals

Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\dfrac{dp(t)}{dt}=\dfrac12p(t)-200$. If $p(0)=100$, then $p(t)$ equals :

If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left(x^2-y^2\right)\dfrac{dy}{dx}=g(x)\,y$, then $g(x)$ equals :

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

The general solution of the differential equation, $\sin2x\left(\dfrac{dy}{dx}-\sqrt{\tan x}\right)-y=0$, is :

If $\dfrac{dy}{dx}+y\tan x=\sin2x$ and $y(0)=1$, then $y(\pi)$ is equal to :