Differential Equations

Let $T$ 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 $T$ at a point $P$ intersect the y-axis at $Y_{P}.$ If $PY_{P}$ has length $1$ for each point $P$ on…

Let $f:(0,\pi)\to R$ be a twice differentiable function such that $limt\to x\frac{f(x)sint-f(t)sinx}{t-x}=sin^{2}x$ for all $x\in(0,\pi)$. If $f(\frac{\pi}{6})=-\frac{\pi}{12}$, then which of the following statement(s)…

Let $f:[0,∞)\to R$ be a continuous function such that $f(x)=1-2x+\int_{0}^{x}e^{x-t}f(t)dt$ for all $x\in[0,∞).$ Then, which of the following statement(s) is (are) TRUE?

Let $y = f(x)$ be the real valued function defined on the interval $(0, \infty)$, satisfying $y(1) = 0$ and the differential equation $x\dfrac{dy}{dx} = y - x^3$. Then which of the following statements is (are) TRUE ?

Let $y : (-\infty, \infty) \rightarrow (0, \infty)$ be the solution of the differential equation $\dfrac{dy}{dx} = \dfrac{e^{5x} y^3 + y^3}{e^x + e^x y^4}$, satisfying $y(0) = \dfrac{1}{\sqrt{2}}$. Then the value of…

Let $f:[1.\infty)\toℝ$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $3\int_{1}^{x}f(t)dt=xf(x)-\frac{x^{3}}{3},x\in[1,\infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$…

For $x\inℝ$, let the function $y(x)$ be the solution of the differential equation $\frac{dy}{dx}+12y=cos(\frac{\pi}{12}x),y(0)=0.$ Then, which of the following statements is/are TRUE?

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha,\beta}(x),x\in R$, be the solution of the differential equation$\frac{dy}{dx}+\alpha y=xe^{\beta x},y(1)=1$Let $S={y_{\alpha,\beta}(x):\alpha,\beta\in R}$. Then…

Let $b$ be a nonzero real number. Suppose $f:ℝ\toℝ$ 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 $x\inℝ,$ then…

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

If $y=y(x)$ satisfies the differential equation $8\sqrt{x}(\sqrt{9+\sqrt{x}})dy=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1}dx$ and $x>0,y(0)=\sqrt{7},$ then $y(256)=$

Let $f:(0,∞)\to R$ be a differentiable function such that $f^{'}(x)=2-\frac{f(x)}{x}\forall x\in(0,\infty)$ and $f(1)\neq1$. Then,