Embibe Experts Solutions for Chapter: Differential Equations, Exercise 2: Exercise-2

Solve: $\frac{xdy}{dx}-\frac{x^3{e}^{{\displaystyle \frac{-y}{x}}}}{2}\left(\pi +2\right)=y+2x^2$

Solution of differential equation $xy\left(mydx+nxdy\right)=\frac{xdy+ydx}{x^m{y}^n}$, given $m+n=1$, is

The solution of differential equation $\left(1-x^2\right)\frac{dy}{dx}+xy=ax$ is

Find the curve which passes through the point $\left(2,0\right)$ such that the segment of the tangent between the point of tangency and the $y$–axis has a constant length equal to $2$ units.

$A$ & $B$ are two separate reservoirs of water. Capacity of reservoir $A$ is double the capacity of reservoir $B$. Both the reservoirs are filled completely with water, their inlet are closed and then the water is released simultaneously from both the reservoirs. The rate of flow of water out of each reservoir at any instant of time is proportional to the quantity of water in the reservoir at that time. One hour after the water is released, the quantity of water in reservoir $A$ is $1.5$ times the quantity of water in reservoir $B$. After how many hours do both the reservoirs have the same quantity of water?

A tank contains $20\mathrm{kg}$ of salt dissolved in $5000 \mathrm{L}$ of water. Brine that contains $0.03\mathrm{kg}$ of salt per litre of water enters the tank at a rate of $25 \mathrm{L}/\min$. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour ?

A curve passing through $\left(1,0\right)$ such that the ratio of the square of the intercept cut by any tangent off the $y$-axis to the subnormal is equal to the ratio of the product of the co-ordinates of the point of tangency to the product of square of the slope of the tangent and the subtangent at the same point. Determine all such possible curves.

The differential equation $\frac{d^{2}y}{d{x}^2}+y+{\cot }^{2}x=0$ must be satisfied by