Differential Equations

Let $y=y(x)$ be the solution curve of the differential equation $\dfrac{dy}{dx}+\dfrac1{x^2-1}y=\left(\dfrac{x-1}{x+1}\right)^{1/2}$, $x>1$ passing through the point $\left(2,\sqrt{\dfrac13}\right)$. Then $\sqrt7\,y(8)$ is equal to :

The differential equation of the family of circles passing through the points (0, 2) and (0, −2) is :

Let the solution curve $y=y(x)$ of the differential equation $\left(1+e^{2x}\right)\left(\dfrac{dy}{dx}+y\right)=1$ pass through the point $\left(0,\dfrac\pi2\right)$. Then, $\displaystyle\lim_{x\to\infty}e^xy(x)$ is equal to :

If the solution curve of the differential equation $\dfrac{dy}{dx}=\dfrac{x+y-2}{x-y}$ passes through the points (2, 1) and (k + 1, 2), k > 0, then

Let $y=y(x)$ be the solution curve of the differential equation $\dfrac{dy}{dx}+\left(\dfrac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y=\dfrac{(x+3)}{x+1}$, x > −1, which passes through the point (0, 1). Then y(1) is equal to :

If $y=y(x)$ satisfies the differential equation $x\,dy+\left(x\tan\left(\dfrac yx\right)-y\right)dx=0$, x > 0, and $y(1)=\dfrac\pi2$, then a value of $y(\sqrt2)$ is

Let the area enclosed by the solution curve $y=y(x)$ of the differential equation $\dfrac{dy}{dx}+\dfrac{x-2}{y-b}=2$, b > 0, y(5) = 0 is 13π. Let $y=y(x)$ intersect the y-axis at the points P and Q. If the tangents to $y=y(x)$ at P and Q meet at T, then the area of ΔPTQ is

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha,\beta}(x)$, $x \in \mathbb R$, be the solution of the differential equation
$$\frac{dy}{dx} + \alpha y = xe^{\beta x},\quad y(1) = 1.$$
Let $S = \{y_{\alpha,\beta}(x) : \alpha, \beta \in \mathbb R\}$. Then which of the following functions belong(s) to the set $S$?

The population P = P(t) at time 't' of a certain species follows the differential equation $\dfrac{dP}{dt}=0.5P-450$. If P(0) = 850, then the time at which population becomes zero is :

Let $b$ be a nonzero real number. Suppose $f:\mathbb R\to\mathbb R$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation

$$f'(x) = \frac{f(x)}{b^2 + x^2}$$

for all $x\in\mathbb R$, then which of the following statements is/are TRUE?

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be functions satisfying

$$f(x+y) = f(x) + f(y) + f(x)f(y)\quad\text{and}\quad f(x) = xg(x)$$

for all $x, y\in\mathbb R$. If $\lim_{x\to0} g(x) = 1$, then which of the following statements is/are TRUE?

Let $\Gamma$ denote a curve $y = y(x)$ which is in the first quadrant and let the point $(1, 0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_P$. If $PY_P$ has length 1 for each point $P$ on $\Gamma$, then which of the following options is/are correct?

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be two non-constant differentiable functions. If

$$f'(x) = \left(e^{(f(x) - g(x))}\right) g'(x) \quad \text{for all } x \in \mathbb{R},$$

and $f(1) = g(2) = 1$, then which of the following statement(s) is (are) TRUE?

Let $f: [0, \infty) \to \mathbb{R}$ be a continuous function such that

$$f(x) = 1 - 2x + \int_0^x e^{x-t} f(t)\, dt$$

for all $x \in [0, \infty)$. Then, which of the following statement(s) is (are) TRUE?

Let $f: (0, \pi) \to \mathbb{R}$ be a twice differentiable function such that

$$\lim_{t \to x} \frac{f(x)\sin t - f(t)\sin x}{t - x} = \sin^2 x \quad \text{for all } x \in (0, \pi).$$

If $f\left(\frac{\pi}{6}\right) = -\frac{\pi}{12}$, then which of the following statement(s) is (are) TRUE?

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with $f(0) = 0$. If $y = f(x)$ satisfies the differential equation

$$\frac{dy}{dx} = (2 + 5y)(5y - 2),$$

then the value of $\displaystyle\lim_{x \to -\infty} f(x)$ is _____.

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with $f(0) = 1$ and satisfying the equation $f(x + y) = f(x)f'(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then, the value of $\log_e(f(4))$ is _____.

If $y = y(x)$ satisfies the differential equation

$$8\sqrt{x}\left(\sqrt{9 + \sqrt{x}}\right)dy = \left(\sqrt{4 + \sqrt{9 + \sqrt{x}}}\right)^{-1}dx, \quad x > 0$$

and $y(0) = \sqrt{7}$, then $y(256) =$

If $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f'(x) > 2f(x)$ for all $x \in \mathbb{R}$, and $f(0) = 1$, then

If $(2+\sin x)\dfrac{dy}{dx}+(y+1)\cos x=0$ and $y(0)=1$, then $y\left(\dfrac\pi2\right)$ is equal to :