Ncert Solutions Maths class 12th

Kindly go through the solution

Kindly go through the solution

The given differential equation is:

$\begin{array}{l}{e}^{x}dy+(y{e}^x+2x)dx=0\\ \Rightarrow e^x\frac{dy}{dx}+y{e}^x+2x=0\\ \Rightarrow \frac{dy}{dx}+y=-2x{e}^{-x}\end{array}$

This is a linear differential equation of the form

$\begin{array}{l}\frac{dy}{dx}+Py=Q,whereP=1\&Q=-2x{e}^{-x}\\ Now,I.F.=e^{{\displaystyle \int Pdx}}=e^{{\displaystyle \int dx}}=e^x\end{array}$

The general solution of the given differential equation is given by,

$\begin{array}{l}y(I.F.)={\displaystyle \int \left(Q\times I.F.\right)dx+C}\\ \Rightarrow y{e}^x={\displaystyle \int (-2x{e}^{-x}.e^x)dx+C}\\ \Rightarrow y{e}^x=-{\displaystyle \int 2xdx+C}\\ \Rightarrow y{e}^x=-x^2+C\\ \Rightarrow y{e}^x+x^2=C\end{array}$

Therefore, option (c) is correct.

The integrating factor of the given differential equation  $\frac{dx}{dy}+P_{1}x=Q_1$

The general solution of the differential equation is given by,

$\begin{array}{l}x(I.F.)={\displaystyle \int \left(Q\times I.F.\right)dy+C}\\ \Rightarrow x.e^{{\displaystyle \int P_{1}dy}}={\displaystyle \int \left(Q_1{e}^{{\displaystyle \int P_{1}dy}}\right)}dy+C\end{array}$

Hence, the correct answer is C.

The given differential equation is:

$\begin{array}{l}\frac{ydx-xdy}{y}=0\\ \Rightarrow \frac{ydx-xdy}{xy}=0\\ \Rightarrow \frac{1}{x}dx-\frac{1}{y}dy=0\end{array}$

Integration both sides, we get:

$\begin{array}{l}log\left|x\right|-log\left|y\right|=logk\\ \Rightarrow log\left|\frac{x}{y}\right|=logk\\ \Rightarrow \frac{x}{y}=k\\ \Rightarrow y=\frac{1}{k}x\\ \Rightarrow y=Cx,where,C=\frac{1}{k}\end{array}$

Therefore, option (C) is correct.

Let the population at any instant (t) be y.

It is given that the rate of increase of population is proportional to the number of inhabitants at any instant.

$\therefore \frac{dy}{dt}\alpha y$

$\Rightarrow \frac{dy}{dt}=ky$ (k is constant)

$\Rightarrow \frac{dy}{y}=kdt$

Integration both sides, we get:

$logy=kt+C..........(1)$

In the year $1999,t=0\&y=20000.$

Therefore, we get:

$log20000=C..........(2)$

In the year $2004,t=5\&y=25000.$

Therefore, we get:

$\begin{array}{l}\Rightarrow log25000=5k+log20000\\ \Rightarrow 5k=log\left(\frac{25000}{20000}\right)=log\left(\frac{5}{4}\right)\\ \Rightarrow k=\frac{1}{5}log\left(\frac{5}{4}\right)..........(3)\end{array}$

In the year&n

$\begin{array}{l}(x+1)\frac{dy}{dx}=2e^{-y}-1\\ \Rightarrow \frac{dy}{2e^{-y}-1}=\frac{dx}{x+1}\\ \Rightarrow \frac{e^{y}dy}{2-e^y}=\frac{dx}{x+1}\end{array}$

Integrating both sides, we get:

$\begin{array}{l}{\displaystyle \int \frac{e^{y}dy}{2-e^y}=log\left|x+1\right|+logC..........(1)}\\ Let2-e^y=t\\ \therefore \frac{d}{dy}(2-e^y)=\frac{dt}{dy}\\ \Rightarrow -e^y=\frac{dt}{dy}\\ \Rightarrow e^{y}dy=-dt\end{array}$

Substituting this value in equation (1), we get:

$\begin{array}{l}{\displaystyle \int \frac{-dt}{t}=log}og\left|x+1\right|+logC\\ \Rightarrow -log\left|t\right|=log\left|C(x+1)\right|\\ \Rightarrow -log\left|2-e^y\right|=log\left|C(x+1)\right|\\ \Rightarrow \frac{1}{2-e^y}=C(x+1)\\ \Rightarrow 2-e^y=\frac{1}{C(x+1)}..........(2)\end{array}$

Now, at x=0& y=0, equation (2) becomes:

$\begin{array}{l}\Rightarrow 2-1=\frac{1}{C}\\ \Rightarrow C=1\end{array}$

Substituting $C=1$ in equation (2), we get:

$\begin{array}{l}2-e^y=\frac{1}{x+1}\\ \Rightarrow e^y=2-\frac{1}{x+1}\\ \Rightarrow e^y=\frac{2x+2-1}{x+1}\\ \Rightarrow e^y=\frac{2x+1}{x+1}\\ \Rightarrow y=log\left|\frac{2x+1}{x+1}\right|,(x\ne -1)\end{array}$

This is the required particular solution of the given differential equation.

The given differential equation is:

$\frac{dy}{dx}+ycotx=4xcosecx$

This equation is a linear equation of the form

$\begin{array}{l}\frac{dy}{dx}+Py=Q,where,p=cotx\&Q=4xcosecx\\ Now,I.F=e^{{\displaystyle \int Pdx}}=e^{{\displaystyle \int cotxdx}}=e^{log\left|sinx\right|}=sinx\end{array}$

The general solution of the given differential equation is given by,

$y(I.F)={\displaystyle \int (Q\times I.F.)dx+C}$

$\begin{array}{l}\Rightarrow ysinx={\displaystyle \int (4xcosecx.sinx)dx+C}\\ \Rightarrow ysinx=4{\displaystyle \int xdx+C}\\ \Rightarrow ysinx=4.\frac{x^2}{2}+C\\ \Rightarrow ysinx=2x^2+C..........(1)\\ Now,y=0at,x=\frac{\pi }{2}\end{array}$

Therefore, equation (1) becomes:

$\begin{array}{l}0=2\times \frac{\pi }{2}+C\\ \Rightarrow C=-\frac{{\pi }^2}{2}\end{array}$

Substituting $C=-\frac{{\pi }^2}{2}$ in equation (1), we get:

$ysinx=2x^2-\frac{{\pi }^2}{2}$

This is the required particular solution of the given differential equa

Kindly go through the solution