Amit M Agarwal Solutions for Chapter: Differential Equations, Exercise 6: Target Exercises

The equation of a curve passing through $(0,1)$ and having gradient $\frac{-\left(y+y^3\right)}{1+x+x{y}^2}$ at $(x, y),$ is

Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops, is proportional to the square root of water depth $y$ where the constant of proportionality $k>0$ depends on the acceleration due to gravity and the geometry of the hole. If $t$ is measured in minutes and $k=\frac{1}{15}$, then the time to drain the tank, if the water is $4 \mathrm{m}$ deep to start with, is

The general solution of the differential equation $y\left(x^{2}y+e^x\right)dx-e^{x}dy=0$ is

The solution of the equation $x^{2}y-x^3\frac{dy}{dx}=y^4\cos x,$ when $y\left(0\right)=1,$ is

A normal is drawn at a point $P(x, y)$ of a curve. It meets the $X$ -axis at $Q .$ If $PQ$ is of constant length $k$, then the differential equation describing such a curve is

Which one of the following curves represents the solution of the initial value problem $Dy=100-y$ where $y\left(0\right)=50 ?$

If a population grows at the rate of $5\%$ per year. Then, the population will be doubled in

The line normal to a given curve at each point $(x, y)$ on the curve passes through the point $(3,0)$. If the curve contains the point $(3,4),$ then its equation is