Differential Equations

The given D.E. is

$\begin{array}{l}\left(1+e^{\frac{x}{y}}\right)dx+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)dy=0\\ \Rightarrow \left(1+e^{\frac{x}{y}}\right)dx=-e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)dy\\ \Rightarrow \frac{dx}{dy}=-\frac{e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)}{1+e^{\frac{x}{y}}}=f\left(\frac{x}{y}\right)\end{array}$

Hence, the given D.E. is homogenous.

Let, $x=yv=\frac{y}{x}$ so that $\frac{dy}{dx}=v+y\frac{dx}{dy}$ in the D.E.

Then, $v+y\frac{dv}{dy}=\frac{-e^v\left(1-v\right)}{1+e^v}$

$\begin{array}{l}\Rightarrow y\frac{dv}{dy}=\frac{v{e}^v-e^v}{1+e^v}-v=\frac{v{e}^v-e^v-v-v{e}^v}{1+e^v}\\ \Rightarrow y\frac{dv}{dy}=-\frac{\left(e^v+v\right)}{1+e^v}\\ \Rightarrow \left(\frac{1+e^v}{e^v+v}\right)dv=-\frac{dy}{y}\end{array}$

Integrating both sides we get,

$log\left|e^v+v\right|=-log\left|y\right|+log\left|c\right|$

Putting back $v=\frac{x}{y}$ we get,

$\begin{array}{l}log\left|e^{\frac{x}{y}}+\frac{x}{y}\right|=log\left|\frac{c}{y}\right|\\ =e^{\frac{x}{y}}+\frac{x}{y}=\frac{c}{y}\end{array}$

$=x+y{e}^{\frac{x}{y}}=c$ is the general solution.

The given D.E is

$\begin{array}{l}ydx+xlog\left(\frac{y}{x}\right).dy-2xdy=0\\ \Rightarrow ydx=\left[2xdy-xlog\left(\frac{y}{x}\right)dy\right]\\ \Rightarrow ydx=\left[2x-xlog\left(\frac{y}{x}\right)\right]dy\\ \Rightarrow \frac{dy}{dx}=\frac{y}{2x-xlog\left(\frac{y}{x}\right)}=\frac{y}{x\left(2-log\frac{y}{x}\right)}\\ =\frac{\frac{y}{x}}{2-log\frac{y}{x}}=f\left(\frac{y}{x}\right)\end{array}$

Hence, the given D.E is homogenous.

Let, $y=vx=\frac{y}{x}=v$ so that $\frac{dy}{dx}=v+x\frac{dv}{dx}$ in the D.E.

Then, $v+x\frac{dv}{dx}=\frac{v}{2-logv}$

$\begin{array}{l}\Rightarrow x\frac{dv}{dx}=\frac{v}{2-logv}-v=\frac{v-2v+vlogv}{2-logv}=\frac{vlogv-v}{2-logv}\\ \Rightarrow \frac{2logv}{v\left[logv-1\right]}dv=\frac{dx}{x}\\ \Rightarrow \frac{1+1-logv}{v\left[logv-1\right]}dv=\frac{dx}{x}\\ \Rightarrow \frac{1-\left[logv-1\right]}{v\left[logv-1\right]}dv=\frac{dx}{x}\end{array}$

$\Rightarrow \left[\frac{1}{v\left(logv-1\right)}-\frac{1}{v}\right]dv=\frac{dx}{x}$

Integrating both sides we get,

$\begin{array}{l}{\displaystyle \int \left[\frac{1}{v\left(logv-1\right)}-\frac{1}{v}\right]dv={\displaystyle \int \frac{dx}{x}}}\\ {\displaystyle \int \frac{dv}{v\left(logv-1\right)}}-logv=logx+logc\end{array}$

Let, $logv-1=t,so,\frac{d}{dv}\left(logv-1\right)=\frac{dt}{dv}$

$\begin{array}{l}\Rightarrow \frac{1}{v}=\frac{dt}{dv}\\ \Rightarrow \frac{dv}{v}=dt\\ \Rightarrow {\displaystyle \int \frac{dt}{t}-logv=logx+logc}\\ \Rightarrow logt-logv=logx+logc\\ \Rightarrow log\left|logv-1\right|-logv=logcx\end{array}$

Putting back $v=\frac{y}{x}$ we get,

$\begin{array}{l}log\left|log\left(\frac{y}{x}\right)-1\right|-log\frac{y}{x}=log\frac{c}{x}\\ =log\left[\frac{log\left(\frac{y}{x}\right)-1}{\frac{y}{x}}\right]=log\frac{c}{x}\\ =\frac{y}{x}\left[log\left(\frac{y}{x}\right)-1\right]=cx\end{array}$

$=log\left(\frac{y}{x}\right)-1=cy$ is the required solution.

The given D.E. is $\frac{xdy}{dx}-y+xsin\left(\frac{y}{x}\right)=0$

$\begin{array}{l}\Rightarrow \frac{xdy}{dx}=y-xsin\left(\frac{y}{x}\right)\\ \Rightarrow \frac{dy}{dx}=\frac{y-xsin\left(\frac{y}{x}\right)}{x}=\frac{y}{x}-sin\frac{y}{x}=f\left(\frac{y}{x}\right)\end{array}$

Hence, the given D.E. is homogenous.

Let, $y=vx=\frac{y}{x}=v$ so that $\frac{dy}{dx}=v+x\frac{dv}{dx}$ in the D.E.

Then, $v+x\frac{dv}{dx}=v-sinv$

$\begin{array}{l}\Rightarrow x\frac{dv}{dx}=-sinv\\ \Rightarrow \frac{dv}{sinv}=-\frac{dx}{x}\\ \Rightarrow cosecvdv=-\frac{dx}{x}\end{array}$

Integrating both sides we get,

$\begin{array}{l}{\displaystyle \int cosecvdv=}{\displaystyle \int -\frac{dx}{x}}\\ \Rightarrow log\left|cosecv-cotv\right|=-logx+logc\\ \Rightarrow log\left|cosecv-cotv\right|=log\frac{c}{x}\\ \Rightarrow cosecv-cotv=\frac{c}{x}\end{array}$

Putting back $v=\frac{y}{x}$ we get,

$\begin{array}{l}cosec\frac{y}{x}-cot\frac{y}{x}=\frac{c}{x}\\ \Rightarrow \frac{1}{sin\frac{y}{x}}-\frac{cos\frac{y}{x}}{sin\frac{y}{x}}=\frac{c}{x}\end{array}$

$\Rightarrow x\left[1-cos\frac{y}{x}\right]=csin\left(\frac{y}{x}\right)$ is the solution of the D.E.

The given D.E is

. $\begin{array}{l}\left\{xcos\left(\frac{y}{x}\right)+ysin\left(\frac{y}{x}\right)\right\}ydx=\left\{ysin\left(\frac{y}{x}\right)-xcos\left(\frac{y}{x}\right)\right\}xdy\\ \Rightarrow \frac{dy}{dx}=\frac{\left\{xcos\frac{y}{x}+ysin\frac{y}{x}\right\}y}{\left\{ysin\frac{y}{x}-xcos\frac{y}{x}\right\}x}=\frac{xycos\frac{y}{x}+y^{2}sin\frac{y}{x}}{xysin\frac{y}{x}-x^{2}cos\frac{y}{x}}\end{array}$

$\Rightarrow \frac{\frac{y}{x}cos\frac{y}{x}+{\left(\frac{y}{x}\right)}^2-sin\frac{y}{x}}{\left(\frac{y}{x}\right)sin\frac{y}{x}-cos\frac{y}{x}}$ {Dividing numerator and denominator by $x^2$ }

$=f\left(\frac{y}{x}\right)$

Hence, the given D.E is homogenous.

Let, $y=vx=\frac{y}{x}=v$ so that $\frac{dy}{dx}=v+x\frac{dv}{dx}$ in the D.E.

Then, $v+x\frac{dv}{dx}=\frac{vcosv+v^{2}sinv}{vsinv-cosv}$

$\begin{array}{l}\Rightarrow x\frac{dv}{dx}=\frac{vcosv+v^{2}sinv}{vsinv-cosv}-v=\frac{vcosv+v^{2}sinv-v^{2}sinv+vcosv}{vsinv-cosv}\\ \Rightarrow \frac{vcosv}{vsinv-cosv}\\ \Rightarrow \left(\frac{vsinv-cosv}{vcosv}\right)dv=\frac{2dx}{x}\end{array}$

Integrating both sides,

$\begin{array}{l}{\displaystyle \int \frac{vsinv-cosv}{vcosv}dv=2{\displaystyle \int \frac{dx}{x}}}\\ \Rightarrow {\displaystyle \int tanvdv-{\displaystyle \int \frac{1}{v}dv=2log\left|x\right|+}}log\left|c\right|\\ \Rightarrow log\left|secv\right|-log\left|v\right|=log{x}^2+logc\\ \Rightarrow log\left|\frac{secv}{v}\right|=logc{x}^2\\ \Rightarrow \frac{secv}{v}=c{x}^2\\ \Rightarrow secv=c{x}^{2}v\end{array}$

Putting back $\Rightarrow v=\frac{y}{x}=c{x}^2\frac{y}{x}=cxy$

$\begin{array}{l}sec\frac{y}{x}=c{x}^2\frac{y}{x}=cxy\\ \Rightarrow \frac{1}{cos\frac{y}{x}}=cxy\\ \Rightarrow \frac{1}{c}=xycos\frac{y}{x}\end{array}$

$\Rightarrow xycos\frac{y}{x}=c_1$ where $c_1=\frac{1}{c}$ 

Kindly go through the solution

Integrating both sides,

The given D.E. is

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

Hence, the D.E. is homogenous $f{x}^n$

Let, $y=vx.=\frac{y}{x}=v$ so that, $\frac{dy}{dx}=v+x\frac{dv}{dx}$ is the D.E.

Thus, $v+x\frac{dv}{dx}=1-2v^2+v$

$\begin{array}{l}\Rightarrow x\frac{dv}{dx}=1-2v^2\\ \Rightarrow \frac{dv}{1-2v^2}=\frac{dx}{x}\end{array}$

Integrating both sides,

The Given D.E. is

$\begin{array}{l}\left(x^2-y^2\right)dx+2xydy=0\\ \Rightarrow 2\times ydy=-\left(x^2-y^2\right)dx\\ =\frac{dy}{dx}=-\frac{\left(x^2-y^2\right)}{2xy}=\frac{\left(y^2-x^2\right)}{2xy}\\ =\frac{1}{2}\frac{\left(\frac{y^2-x^2}{x^2}\right)}{\left(\frac{xy}{x^2}\right)}\\ =\frac{1}{2}\frac{\left[{\left(\frac{y}{x}\right)}^2-1\right]}{\left(\frac{y}{x}\right)}=f\left(\frac{y}{x}\right)\end{array}$

Hence, the given D.E. is homogenous.

Let, $y=vx\Rightarrow \frac{y}{x}=v,sothat,\frac{dy}{dx}=v+x\frac{dv}{dx}$ in the D.E

$\begin{array}{l}\Rightarrow v+x\frac{dv}{dx}=\frac{1}{2}\left[\frac{v^2-1}{v}\right]\\ \Rightarrow x\frac{dv}{dx}=\frac{v^2-1}{2v}-v=\frac{v^2-1-2v^2}{2v}=\frac{-1-v^2}{2v}\\ \Rightarrow \left(\frac{2v}{1+v^2}\right)dv=-\frac{dx}{x}\end{array}$

Integrating both sides we get,

Putting back $v=\frac{y}{x}$ we get,

$\begin{array}{l}\Rightarrow x\left[\frac{y^2}{x^2}+1\right]=c\\ \Rightarrow \frac{y^2+x^2}{x}=c\end{array}$

$\Rightarrow y^2+x^2=xc$ is the required solution.

The Given D.E. is $\left(x-y\right)dy-\left(x+y\right)dx=0$

$\begin{array}{l}\Rightarrow \left(x-y\right)dy=\left(x+y\right)dx\\ \Rightarrow \frac{dy}{dx}=\frac{x+y}{x-y}=\frac{x\left(1+\frac{y}{x}\right)}{x\left(1-\frac{y}{x}\right)}=\frac{1+\frac{y}{x}}{1-\frac{y}{x}}=f\left(\frac{y}{x}\right)\end{array}$

Hence, the given D.E. is homogenous.

Let, $y=vx=\frac{y}{x}=v,so,\frac{dy}{dx}=v+x\frac{dv}{dx}$ in the D.E

Then, $\begin{array}{l}v+x\frac{dv}{dx}=\frac{1+v}{1-v}\\ \Rightarrow x\frac{dv}{dx}=\frac{1+v}{1-v}-v=\frac{1+v-1+v^2}{1-v}=\frac{1+v^2}{1-v}\\ \Rightarrow \left[\frac{1-v}{1+v^2}\right]dv=\frac{dx}{x}\end{array}$

Integrating both sides,

$\begin{array}{l}{\displaystyle \int \frac{1-v}{1+v^2}dv={\displaystyle \int \frac{dx}{x}}}\\ \Rightarrow {\displaystyle \int \frac{1}{1+v^2}dv-\frac{1}{2}{\displaystyle \int \frac{2v}{1+v^2}dv=logx+c}}\\ \Rightarrow ta{n}^{-1}v-\frac{1}{2}log\left(1+v^2\right)=logx+c\end{array}$

Putting back $v=\frac{y}{x},weget,$

$\begin{array}{l}=ta{n}^{-1}\frac{y}{x}-\frac{1}{2}log\left|1+\frac{y^2}{x^2}\right|=logx+c\\ =ta{n}^{-1}\frac{y}{x}-\frac{1}{2}log\left|\frac{x^2+y^2}{x^2}\right|=logx+c\\ =ta{n}^{-1}\frac{y}{x}-\frac{1}{2}\left[log\left(x^2+y^2\right)-log{x}^2\right]=logx+c\end{array}$

$\begin{array}{l}=ta{n}^{-1}\frac{y}{x}-\frac{1}{2}log\left(x^2+y^2\right)+\frac{1}{2}log{x}^2=logx+c\\ =ta{n}^{-1}\frac{y}{x}-\frac{1}{2}log\left(x^2+y^2\right)+log{\left(x^2\right)}^{\frac{1}{2}}=logx+c\\ =ta{n}^{-1}\frac{y}{x}-\frac{1}{2}log\left(x^2+y^2\right)+logx=log+c\\ =ta{n}^{-1}\frac{y}{x}=\frac{1}{2}log\left(x^2+y^2\right)+c\end{array}$

The given D.E. is

$\begin{array}{l}{y}^1=\frac{x+y}{x}\\ =\frac{dy}{dx}=1+\frac{y}{x}=f\left(\frac{y}{x}\right)\end{array}$

Hence, the given D.E is homogenous.

Let,  $y=vx\to \frac{y}{x}=v,sothat,\frac{dy}{dx}=v+\frac{xdv}{dx}$

So, the D.E. becomes

$\begin{array}{l}v+\frac{xdv}{dx}=1+v\\ \Rightarrow \frac{xdv}{dx}=1\\ \Rightarrow dv=\frac{dx}{x}\end{array}$

Integrating both sides,

$\begin{array}{l}{\displaystyle \int dv={\displaystyle \int \frac{dx}{x}}}\\ v=log\left|x\right|+c\end{array}$

Putting $v=\frac{y}{x}$ back we get,

$\frac{y}{x}=log\left|x\right|+c$

The given D.E. is

$\begin{array}{l}\frac{dy}{dx}=\frac{x^2+y^2}{x^2+x+y}=\frac{x^2\left(1+\frac{y^2}{x^2}\right)}{x^2\left(1+\frac{y}{x}\right)}=F\left(x,y\right)\\ Then,F\left(\lambda x,\lambda y\right)=\frac{{\lambda }^2{x}^2}{{\lambda }^2{x}^2}\left[\frac{1+\frac{{\lambda }^2{y}^2}{{\lambda }^2{x}^2}}{1+\frac{\lambda y}{\lambda x}}\right]={\lambda }^{0}F\left(x,y\right)\\ ={\lambda }^{2}F\left(x,y\right)\end{array}$

Hence, $F\left(x,y\right)$ is a homogenous $f{x}^n$ of degree 2.

To solve it, let

$\begin{array}{l}y=vx,sothat,\frac{dy}{dx}=v.\frac{dx}{dx}+x\frac{dv}{dx}\\ v=\frac{y}{x}\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}\end{array}$

The D.E. now becomes,

$\begin{array}{l}v+x\frac{dv}{dx}=\frac{1+v^2}{1+v}\\ \Rightarrow x\frac{dv}{dx}=\frac{1+v^2}{1+v}-v=\frac{1+v^2-v-v^2}{1+v}=\frac{1-v}{1+v}\\ \Rightarrow \left(\frac{1+v}{1-v}\right)dv=\frac{dx}{x}\end{array}$

Integrating both sides,

$\begin{array}{l}{\displaystyle \int \frac{1+v}{1-v}dv={\displaystyle \int \frac{dx}{x}}}\\ \Rightarrow {\displaystyle \int \frac{1+v}{1-v}dv=logx+c}\\ \Rightarrow {\displaystyle \int \frac{2-\left(1-v\right)}{1-v}dv=logx+c}\\ \Rightarrow {\displaystyle \int \frac{2}{1-v}dv-{\displaystyle \int 1dv=logx+c}}\\ \Rightarrow \frac{2log\left|1-v\right|}{-1}-v=logx+c\\ \Rightarrow log{\left(1-v\right)}^2+v=-logx-c\end{array}$

Put $v=\frac{y}{x},$

$\begin{array}{l}log{\left(1-\frac{y}{x}\right)}^2+\frac{y}{2}=-logx+c\\ \Rightarrow log{\left(\frac{x-y}{x}\right)}^2+logx=-\frac{y}{x}-c\end{array}$

$\begin{array}{l}\Rightarrow log\left[{\left(\frac{x-y}{x}\right)}^2\times x\right]=-\frac{y}{x}-c\\ \Rightarrow \frac{x-y}{x}=e^{-\frac{y}{x}-c}=c_1{e}^{-\frac{y}{x}-c},where,c_{1}e\end{array}$

$\Rightarrow {\left(x-y\right)}^2=c_1.x{e}^{-\frac{y}{x}}$ is the required solution of the D.E.