D.P. GUPTA, Preetima Bajpai and, Sanjeev Kumar Jha Solutions for Chapter: Differential Equations, Exercise 2: Exercise 2

In a certain chemical reaction the rate of conversion of a substance at time $t$ is proportional to the quantity of the substance still untransformed at that instant. At the end of one hour, $60 \mathrm{gms}$ remain and at the end of $4$ hours $21 \mathrm{gms}$ remain. How many grams of the substance was there initially? $\left[{\left(\frac{{60}^4}{21}\right)}^{1/3}=85.15\right]$

A drug is excreted in a patients urine. The urine is monitored continuously using a catheter. A patient is administered $10 \mathrm{mg}$ of drug at time $t=0$, which is excreted at a rate of $-3t^{\frac{1}{2}} \mathrm{mg}/\mathrm{h}$
$\left(\mathrm{i}\right)$ What is the general equation for the amount of drug in the patient at time $t>0$?
$\left(\mathrm{ii}\right)$ When will the patient be drug free?

The tangent at a point $P\left(x\mathit{,}y\right)$ on a curve meets the axes at $P_1$ and $P_2$ such that $P$ divides $P_1{P}_2$ internally in the ratio $2:1$. The equation of the curve is

The temperature $T$ of a cooling object drops at a rate proportional to the difference $T\mathit{-}S\mathit{,}$ where $S$ is constant temperature of surrounding medium. If initially $T={150}^{°}C$, find the temperature of the cooling object at any time $t$

A bank pays interest by continuous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at $8\%$ per year compounded continuously. Calculate the percentage increase in such an account over one year [Take $e^{08}=1·0833$]

If $y+\frac{d}{dx}\left(xy\right)=x(\sin x+\log x)$ then

The solution of $\frac{d^{2}x}{d{y}^2}-x=k,$ where $k$ is a non-zero constant, vanishes when $y=0$ and tends of finite limit as $y$ tends to infinity, is

The general solution of the differential equation $\frac{dy}{dx}+y g^{\text{'}}\left(x\right)=g\left(x\right)·g^{\text{'}}\left(x\right)$ where $g\left(x\right)$ is a given function of $x$ is