Class 12th

105. Given,  $y=5cosx-3sinx$

Differentiating w r t x we get,

$\frac{dy}{dx}=5\frac{d}{dx}cosx-3\frac{d}{dx}sinx$

$-5sinx-3cosx.$

Differentiating again w r t. 'x' we get,

$\frac{d^{2}y}{d{x}^2}=-5\frac{d}{dx}sinx-3\frac{d}{dx}cosx$

$=-5cosx+3sinx$

$=-\left[5cosx-3sinx\right]$

$=-y$

$\Rightarrow \frac{d^{2}y}{d{x}^2}+y=0$ . Hence proved.

Given,  $x^5\frac{dy}{dx}=-y^5$

$\Rightarrow \frac{dy}{y^5}=-\frac{dx}{x^5}$

Integrating both sides

$\begin{array}{l}{\displaystyle \int \frac{dy}{y^5}=-{\displaystyle \int \frac{dx}{x^5}}}\\ \Rightarrow {\displaystyle \int y^{-5}dy=-{\displaystyle \int x^{-5}dx}}\end{array}$

$\begin{array}{l}\Rightarrow \frac{y^{-5+1}}{\left(-5+1\right)}=-\frac{x^{-5+1}}{\left(-5+1\right)}+c\\ \Rightarrow \frac{1}{-4y^4}=\frac{1}{4x^4}+c\end{array}$

$\Rightarrow \frac{1}{y^4}=\frac{1}{x^4}+4c$ is the general solution.

Given, $ylogydx-xdy=0$

$\begin{array}{l}\Rightarrow ylogydx=xdy\\ \Rightarrow \frac{dy}{ylogy}=\frac{dx}{x}\end{array}$

Integration both sides,

${\displaystyle \int \frac{dy}{ylogy}={\displaystyle \int \frac{dx}{x}}}$

Put log $y=t\Rightarrow \frac{1}{y}=\frac{dt}{dy}\Rightarrow \frac{dy}{y}=dt$

Hence, ${\displaystyle \int \frac{dt}{t}={\displaystyle \int \frac{dx}{x}}}$

$\begin{array}{l}\Rightarrow log\left|t\right|=log\left|x\right|+log\left|c\right|=log\left|xc\right|\\ \Rightarrow t=\pm xc\end{array}$

$\Rightarrow logy=ax$ where $a=\pm c$

$\Rightarrow y=e^{ax}$ is the general solution.

104. Let $y=sin\left(logx\right)$

so, $\frac{dy}{dx}=\frac{d}{dx}sin\left(logx\right)=cos\left(logx\right)\frac{d}{dx}logx=\frac{cos\left(logx\right)}{x}$

$\therefore \frac{d^{2}y}{d{x}^2}=\frac{x\frac{d}{dx}cos\left(logx\right)-cos\left(logx\right)\frac{dx}{dx}}{x^2}$

$=\frac{x\left[-sin\left(logx\right)\right]\frac{d}{dx}logx-cos\left(logx\right)}{x^2}$

$=\frac{-\left[x\cdot sin\left(logx\right)*\frac{1}{x}+cos\left(logx\right)\right]}{x^2}$

$=\frac{-\left[sin\left(logx\right)+cos\left(logx\right)\right]}{x^2}$

Given,  $\frac{dy}{dx}=\left(1+x^2\right)\left(1+y^2\right)$

$\Rightarrow \frac{dy}{\left(1+y^2\right)}=\left(1+x^2\right)dx$

Integrating both sides

$\begin{array}{l}{\displaystyle \int \frac{dy}{\left(1+y^2\right)}dy={\displaystyle \int \left(x^2+1\right)dx}}\\ \Rightarrow ta{n}^{-1}\frac{y}{1}=\frac{x^3}{3}+x+c\end{array}$

$\Rightarrow ta{n}^{-1}y=\frac{x^3}{3}+x+c$ is the general solution.

103. Let $y=log\left(logx\right)$

So,  $\frac{dy}{dx}=\frac{1}{logx}\frac{d}{dx}logx=\frac{1}{xlogx}$

$\therefore \frac{d^{2}y}{d{x}^2}=\frac{xlogx\frac{d}{dx}\left(1\right)-1\cdot \frac{d}{dx}\left(xlogx\right)}{{\left(xlogx\right)}^2}$

$=\frac{-\left[x\frac{d}{dx}logx+logx\frac{dx}{dx}\right]}{{\left[xlogx\right]}^2}$

$=\frac{-\left(x*\frac{1}{x}+logx\right)}{{\left[xlogx\right]}^2}$

$=\frac{-\left(1+logx\right)}{{\left(xlogx\right)}^2}$

Given,  $\left(e^x+e^{-x}\right)dy-\left(e^x-e^{-x}\right)dx=0$

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

Integrating both sides

$\Rightarrow {\displaystyle \int dy=}\frac{e^x-e^{-x}}{e^x+e^{-x}}dx\left\{?{\displaystyle \int \frac{f^{|}\left(x\right)}{f\left(x\right)}}dx=log\left|x\right|\right\}$

$\Rightarrow y=log\left|e^x+e^{-x}\right|+c$ is the required general solution.

Given, $se{c}^{2}xtanydx+se{c}^{2}ytanxdy=0$

Dividing throughout by ' $tanxtany$ ' we get,

$\begin{array}{l}\frac{se{c}^{2}xtany}{tanxtany}dx+\frac{se{c}^{2}ytanx}{tanxtany}dy=0\\ \Rightarrow \frac{se{c}^{2}x}{tanx}dx+\frac{se{c}^{2}y}{tany}dy=0\end{array}$

Integrating both sides we get,

$\begin{array}{l}{\displaystyle \int \frac{se{c}^{2}x}{tanx}dx+{\displaystyle \int \frac{se{c}^{2}y}{tany}dy=logc}}\\ \Rightarrow log\left|tanx\right|+log\left|tany\right|=logc\left\{{\displaystyle \int \frac{f^{|}\left(x\right)}{f\left(x\right)}}dx-log\left|f\left(x\right)\right|\right\}\\ \Rightarrow log\left|\left(tanx+tany\right)\right|=logc\end{array}$

$\Rightarrow tanxtany=\pm c$ is the required general solution.

102. Let $y=ta{n}^{-1}x$

So,  $\frac{dy}{dx}=\frac{d}{dx}ta{n}^{-1}x=\frac{1}{1+x^2}$

$\therefore \frac{d^{2}y}{d{x}^2}=\frac{\left(1+x^2\right)\frac{d}{dx}\left(1\right)-\left(1\right)\frac{d}{dx}\left(1+x^2\right)}{{\left(1+x^2\right)}^2}$

$=\frac{-2x}{{\left(1+x^2\right)}^2}$

Given,  $\frac{dy}{dx}+y=1$

$\Rightarrow \frac{dy}{dx}=1-y=-\left(y-1\right)$

By separable of variable,

$\frac{dy}{\left(y-1\right)}=-dx$

Integrating both sides,

$\begin{array}{l}{\displaystyle \int \frac{dy}{\left(y-1\right)}=-{\displaystyle \int dx}}\\ \Rightarrow log\left|y-1\right|=-x+c\\ \Rightarrow \left|y-1\right|=e^{-x+c}\\ \Rightarrow y-1=\pm e^{-x}.e^c\end{array}$

$\Rightarrow y=1+Ac$ where $A=\pm e^c$

Is the general solution.