Embibe Experts Solutions for Chapter: Differential Equations, Exercise 1: Exercise 1

Solution of differential equation $\frac{x+y\frac{dy}{dx}}{y-x\frac{dy}{dx}}=\frac{x {\cos }^2\left(x^2+y^2\right)}{y^3}$ is equal to

The function $f\left(x\right)$ satisfying the equation, $f^2\left(x\right)+4f^{\text{'}}\left(x\right)·f\left(x\right)+{\left[f^{\text{'}}\left(x\right)\right]}^2=0$ is -

A normal is drawn at a point $P(x,y)$ of a curve. It meets the $x$-axis and the $y$-axis in point $A$ and $B,$ respectively, such that $\frac{1}{OA}+\frac{1}{OB}=1,$ where $O$ is the origin, the equation of such a curve is a circle which passes through $\left(5,4\right)$ and has -

Let $y=y\left(x\right)$ be the solution of the differential equation $xdy-ydx=\sqrt{\left(x^2-y^2\right)}dx, x\ge 1$, with $y\left(1\right)=0$. If the area bounded by the line $x=1,x=e^{\pi },y=0$ and $y=y\left(x\right)$ is $\alpha e^{2\pi }+\beta ,$ then the value of $10(\alpha +\beta )$ is equal to ___ .

Let $f:R\to R$ be a differentiable function with $f\left(0\right)=0$. If $y=f\left(x\right)$ satisfies the differential equation $\frac{dy}{dx}=\left(2+5y\right)\left(5y-2\right)$, then the value of $\underset{x\to -\infty }{\lim }f\left(x\right)$ is _____ .

Let $f\left(x\right)$ be a continuous and differentiable function in $x\in \left[0,\frac{1}{4}\right]$ such that $f\left(0\right)=0$ and $f^{\text{'}}\left(x\right)-{\left\{f\left(x\right)\right\}}^2=1$, then number of solutions of equation $f\left(x\right)=1$ where $x\in \left[0,\frac{1}{4}\right]$

The function $y=f\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-4}=\frac{7x^6+2x}{\sqrt{4-x^2}}$ in $\left(-2,2\right)$ satisfying $f\left(0\right)=1.$ If ${\int }_{-\sqrt{3}}^{\sqrt{3}}f\left(x\right)dx=a\pi +b\sqrt{3},a,b$ are rational numbers, then value of $3a+2b$ is

If function $y=f\left(x\right)$ satisfy the differential equation $\left(x^3+1\right)dy=x\left(1-3xy\right)dx$ and $f\left(0\right)=0,$ then $\underset{x\to 0}{\lim }\left\{\frac{x^2}{f\left(x\right)}\right\}$ is equal to