Differential Equations

D.E. $(1 + {e^{2x}})\left( {{{dy} \over {dx}} + y} \right) = 1$

$ \Rightarrow {{dy} \over {dx}} + y = {1 \over {1 + {e^{2x}}}}$

I.F. $ = {e^{\int {1\,.\,dx} }} = {e^x}$

$\therefore$ Solution

${e^x}y(x) = \int {{{{e^x}} \over {1 + {e^{2x}}}}dx} $

$ \Rightarrow {e^x}y(x) = {\tan ^{ - 1}}({e^x}) + C$

$\because$ It passes through $\left( {0,{\pi \over 2}} \right),\,C = {\pi \over 2} - {\pi \over 4} = {\pi \over 4}$

$\therefore$ $\mathop {\lim }\limits_{x \to \infty } {e^x}y(x) = \mathop {\lim }\limits_{x \to \infty } {\tan ^{ - 1}}({e^x}) + {\pi \over 4}$

$ = {{3\pi } \over 4}$

${{dy} \over {dx}} + {1 \over {{x^2} - 1}}y = \sqrt {{{x - 1} \over {x + 1}}} ,\,x > 1$

Integrating factor I.F. $ = {e^{\int {{1 \over {{x^2} - 1}}dx} }} = {e^{{1 \over 2}\ln \left| {{{x - 1} \over {x + 1}}} \right|}}$

$ = \sqrt {{{x - 1} \over {x + 1}}} $

Solution of differential equation

$y\sqrt {{{x - 1} \over {x + 1}}} = \int {{{x - 1} \over {x + 1}}dx = \int {\left( {1 - {2 \over {x + 1}}} \right)dx} } $

$y\sqrt {{{x - 1} \over {x + 1}}} = x - 2\ln |x + 1| + C$

Curve passes through $\left( {2,\sqrt {{1 \over 3}} } \right)$

${1 \over {\sqrt 3 }} \times {1 \over {\sqrt 3 }} = 2 - 2\ln 3 + C$

$C = 2\ln 3 - {5 \over 3}$

$y(8) \times {{\sqrt 7 } \over 3} = 8 - 2\ln 9 + 2\ln 3 - {5 \over 3}$

$\sqrt 7 \,.\,y(8) = 19 - 6\ln 3$

Family of circles passing through the points (0, 2) and (0, $-$2)

${x^2} + (y - 2)(y + 2) + \lambda x = 0,\,\lambda \in R$

${x^2} + {y^2} + \lambda x - 4 = 0$ ...... (1)

Differentiate w.r.t x

$2x + 2y{{dy} \over {dx}} + \lambda = 0$ ....... (2)

Using (1) and (2), eliminate $\lambda$

${x^2} + {y^2} - \left( {2x + 2y{{dy} \over {dx}}} \right)x - 4 = 0$

$2xy{{dy} \over {dx}} + {x^2} - {y^2} + 4 = 0$

${{xdy - ydx} \over {\sqrt {{x^2} + {y^2}} }} = dx$

$ \Rightarrow {{dy} \over {dx}} = {{\sqrt {{x^2} + {y^2}} } \over x} + {y \over x}$

$ \Rightarrow {{dy} \over {dx}} = \sqrt {1 + {{{y^2}} \over {{x^2}}}} + {y \over x}$

Let ${y \over x} = v$

$ \Rightarrow v + x{{dv} \over {dx}} = \sqrt {1 + {v^2}} + v$

$ \Rightarrow {{dv} \over {\sqrt {1 + {v^2}} }} = {{dx} \over x}$

OR $\ln \left( {v + \sqrt {1 + {v^2}} } \right) = \ln x + C$

at $x = 1,\,y = 0$

$ \Rightarrow C = 0$

${y \over x} + \sqrt {1 + {{{y^2}} \over {{x^2}}}} = x$

At $x = 2$,

${y \over 2} + \sqrt {1 + {{{y^2}} \over 4}} = 2$

$ \Rightarrow 1 + {{{y^2}} \over 4} = 4 + {{{y^2}} \over 4} - 2y$

OR $y = {3 \over 2}$

$({\sin ^2}2x){{dy} \over {dx}} + (8{\sin ^2}2x + 2\sin 4x)y$

$ = 2{e^{ - 4x}}(2\sin 2x + \cos 2x)$

${{dy} \over {dx}} + (8 + 4\cot 2x)y = 2{e^{ - 4x}}\left( {{{2\sin 2x + \cos 2x} \over {{{\sin }^2}2x}}} \right)$

Integrating factor

$(I.F.) = {e^{\int {(8 + 4\cot 2x)dx} }}$

$ = {e^{8x + 2\ln \sin 2x}}$

Solution of differential equation

$y.\,{e^{8x + 2\ln \sin 2x}}$

$ = \int {2{e^{(4x + 2\ln \sin 2x)}}{{(2\sin 2x + \cos 2x)} \over {{{\sin }^2}2x}}dx} $

$ = 2\int {{e^{4x}}(2\sin 2x + \cos 2x)dx} $

$y.\,{e^{8x + 2\ln \sin 2x}} = {e^{4x}}\sin 2x + c$

$y\left( {{\pi \over 4}} \right) = {e^{ - \pi }}$

${e^{ - \pi }}\,.\,{e^{2\pi }} = {e^\pi } + c \Rightarrow c = 0$

$y\left( {{\pi \over 6}} \right) = {{{e^{{{2\pi } \over 3}}}{{\sqrt 3 } \over 2}} \over {{e^{\left( {{{4\pi } \over 3} + 2\ln {{\sqrt 3 } \over 2}} \right)}}}}$

$ = {e^{{{ - 2\pi } \over 3}}}\,.\,{2 \over {\sqrt 3 }}$

${{dy} \over {dx}} = x + y$

Let $x + y = t$

$1 + {{dy} \over {dx}} = {{dt} \over {dx}}$

${{dt} \over {dx}} - 1 = t \Rightarrow \int {{{dt} \over {t + 1}} = \int {dx} } $

$\ln |t + 1| = x + C'$

$|t + 1| = C{e^x}$

$|x + y + 1| = C{e^x}$

For ${y_1}(x),\,{y_1}(0) = 0 \Rightarrow C = 1$

For ${y_2}(x),\,{y_2}(0) = 1 \Rightarrow C = 2$

${y_1}(x)$ is given by $|x + y + 1| = {e^x}$

${y_2}(x)$ is given by $|x + y + 1| = 2{e^x}$

At point of intersection

${e^x} = 2{e^x}$

No solution

So, there is no point of intersection of ${y_1}(x)$ and ${y_2}(x)$.

${{dy} \over {dx}} + {{xy} \over {{x^2} - 1}} = {{{x^4} + 2x} \over {\sqrt {1 - {x^2}} }}$

which is first order linear differential equation.

Integrating factor $(I.F.) = {e^{\int {{x \over {{x^2} - 1}}dx} }}$

$ = {e^{{1 \over 2}\ln |{x^2} - 1|}} = \sqrt {|{x^2} - 1|} $

$ = \sqrt {1 - {x^2}} $

$\because$ $x \in ( - 1,1)$

Solution of differential equation

$y\sqrt {1 - {x^2}} = \int {({x^4} + 2x)dx = {{{x^5}} \over 5} + {x^2} + c} $

Curve is passing through origin, $c = 0$

$y = {{{x^5} + 5{x^2}} \over {5\sqrt {1 - {x^2}} }}$

$\int\limits_{{{ - \sqrt 3 } \over 2}}^{{{\sqrt 3 } \over 2}} {{{{x^5} + 5{x^2}} \over {5\sqrt {1 - {x^2}} }}dx = 0 + 2\int\limits_0^{{{\sqrt 3 } \over 2}} {{{{x^2}} \over {\sqrt {1 - {x^2}} }}dx} } $

put $x = \sin \theta $

$dx = \cos \theta \,d\theta $

$I = 2\int\limits_0^{{\pi \over 3}} {{{{{\sin }^2}\theta \,.\,\cos \theta d\theta } \over {\cos \theta }}} $

$ = \int\limits_0^{{\pi \over 3}} {(1 - \cos 2\theta )d\theta } $

$ = \left. {\left( {\theta - {{\sin 2\theta } \over 2}} \right)} \right|_0^{{\pi \over 3}}$

$ = {\pi \over 3} - {{\sqrt 3 } \over 4}$

${{dy} \over {dx}} + 2y\tan x = \sin x$

which is a first order linear differential equation.

Integrating factor (I. F.) $ = {e^{\int {2\tan x\,dx} }}$

$ = {e^{2\ln |\sec x|}} = {\sec ^2}x$

Solution of differential equation can be written as

$y\,.\,{\sec ^2}x = \int {\sin x\,.\,{{\sec }^2}x\,dx = \int {\sec \,x\,.\,\tan x\,dx} } $

$y~{\sec ^2}x = \sec x + C$

$y\left( {{\pi \over 3}} \right) = 0,0 = \sec {\pi \over 3} + C \Rightarrow \,\,\,\,C = - 2$

$y = {{\sec x - 2} \over {{{\sec }^2}x}} = \cos x - 2{\cos ^2}x$

$ = {1 \over 8} - 2{\left( {\cos x - {1 \over 4}} \right)^2}$

${y_{\max }} = {1 \over 8}$

${{dy} \over {dx}} \propto {{ - y} \over x}$

${{dy} \over {dx}} = {{ - ky} \over x} \Rightarrow \int {{{dy} \over y} = - K\int {{{dx} \over x}} } $

$\ln |y| = - K\ln |x| + C$

If the above equation satisfy (1, 2) and (8, 1)

$\ln 2 = - K \times 0 + C \Rightarrow C = \ln 2$

$\ln 1 = - K\ln 8 + \ln 2 \Rightarrow K = {1 \over 3}$

So, at $x = {1 \over 8}$

$\ln |y| = - {1 \over 3}\ln \left( {{1 \over 8}} \right) + \ln 2 = 2\ln 2$

$|y| = 4$

Given,

${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$

$ \Rightarrow bxdy + cydy + ady = axdx - bydx + adx$

$ \Rightarrow bxdy + bydx + (cy + a)dy = (ax + a)dx$

$ \Rightarrow b(xdy + ydx) + (cy + a)dy = (ax + a)dx$

$ \Rightarrow bd(xy) + (cy + a)dy = (ax + a)dx$

Integrating both sides, we get

$ \Rightarrow b\int {d(xy) + \int {(cy + a)dy = \int {(ax + a)dx} } } $

$ \Rightarrow b\,.\,xy + c\,.\,{{{y^2}} \over 2} + ay = {{a{x^2}} \over 2} + ax + k$

$ \Rightarrow {{a{x^2}} \over 2} - {{c{y^2}} \over 2} - bxy + ax - ay + k = 0$

For equation of circle,

Coefficient of x² = Coefficient of y²

$\therefore$ ${a \over 2} = - {c \over 2}$

$ \Rightarrow a = - c$

And coefficient of $xy = 0$

$\therefore$ $ - b = 0$

$ \Rightarrow b = 0$

$\therefore$ Circle equation becomes,

${{a{x^2}} \over 2} + {{a{y^2}} \over 2} + ax - ay + k = 0$

$ \Rightarrow {x^2} + {y^2} + 2x - 2y + {{2k} \over a} = 0$

$\therefore$ Center $ = ( - g, - f) = ( - 1,1) = (\alpha ,\beta )$

$\therefore$ $\alpha = - 1$ and $\beta = 1$

$\therefore$ $\alpha + 2\beta = - 1 + 2 \times (1) = 1$

Given,

$(1 + {e^{2x}}){{dy} \over {dx}} + 2(1 + {y^2}){e^x} = 0$

$ \Rightarrow {{dy} \over {dx}} = {{ - 2(1 + {y^2}){e^x}} \over {1 + {e^{2x}}}}$

$ \Rightarrow \int {{{dy} \over {1 + {y^2}}} = \int {{{ - 2{e^x}dx} \over {(1 + {e^{2x}})}}} } $

$ \Rightarrow {\tan ^{ - 1}}(y) = \int {{{ - 2{e^x}dx} \over {(1 + {e^{2x}})}}} $

Now,

Let ${e^x} = t$

$ \Rightarrow {e^x}dx = dt$

$ \Rightarrow {\tan ^{ - 1}}(y) = \int {{{ - 2dt} \over {1 + {t^2}}}} $

$ \Rightarrow {\tan ^{ - 1}}(y) = - 2{\tan ^{ - 1}}(t) + C$

$ \Rightarrow {\tan ^{ - 1}}(y) = - 2{\tan ^{ - 1}}({e^x}) + C$ [Putting value of t]

Given, y(0) = 0 means when x = 0 the y = 0

$\therefore$ ${\tan ^{ - 1}}(0) = - 2{\tan ^{ - 1}}({e^o}) + C$

$ \Rightarrow 0 = - 2 \times {\pi \over 4} + C$

$ \Rightarrow C = {\pi \over 2}$

$\therefore$ ${\tan ^{ - 1}}(y) = 2{\tan ^{ - 1}}({e^x}) + {\pi \over 2}$

$ \Rightarrow y = \tan \left( { - 2{{\tan }^{ - 1}}({e^x}) + {\pi \over 2}} \right)$

Differentiating both sides, we get

$y' = {\sec ^2}\left( { - 2{{\tan }^{ - 1}}({e^x}) + {\pi \over 2}} \right) - 2\,.\,{{{e^x}} \over {1 + {e^{2x}}}}$

$ = {\sec ^2}\left( { - 2{{\tan }^{ - 1}}({e^x}) + {\pi \over 2}} \right) \times {{ - 2{e^x}} \over {1 + {e^{2x}}}}$

$\therefore$ $y'(0) = {\sec ^2}\left( { - 2{{\tan }^{ - 1}}({e^o}) + {\pi \over 2}} \right) \times {{ - 2{e^o}} \over {1 + {e^o}}}$

$ = {\sec ^2}\left( { - 2 \times {\pi \over 4} + {\pi \over 2}} \right) \times {{ - 2} \over 2}$

$ = {\sec ^2}(0)x - 1$

$ = 1 \times 1$

$ = - 1$

And

$y\left( {\log _e^{\sqrt 3 }} \right) = \tan \left( { - 2{{\tan }^{ - 1}}\left( {{e^{\log _e^{\sqrt 3 }}}} \right) + {\pi \over 2}} \right)$

$ = \tan \left( { - 2{{\tan }^{ - 1}}(\sqrt 3 ) + {\pi \over 2}} \right)$

$ = \tan \left( { - 2 \times {\pi \over 3} + {\pi \over 2}} \right)$

$ = \tan \left( { - {\pi \over 6}} \right)$

$ = - \tan \left( {{\pi \over 6}} \right)$

$ = - {1 \over {\sqrt 3 }}$

$\therefore$ $6\left( {y'(0) + {{\left( {y(\log _e^{\sqrt 3 })} \right)}^2}} \right)$

$ = 6\left( { - 1 + {{\left( {{{ - 1} \over {\sqrt 3 }}} \right)}^2}} \right)$

$ = 6\left( { - 1 + {1 \over 3}} \right) = 6 \times {{ - 2} \over 3} = - 4$

Given,

$x{{dy} \over {dx}} - y = \sqrt {{y^2} + 16x} $

$ \Rightarrow x{{dy} \over {dx}} = y + \sqrt {{y^2} + 16x} $

$ \Rightarrow {{dy} \over {dx}} = {y \over x} + \sqrt {{{\left( {{y \over x}} \right)}^2} + 16} $

This is a homogenous different equation.

Let ${y \over x} = v$

$ \Rightarrow y = vx$

$ \Rightarrow {{dy} \over {dx}} = v + x{{dv} \over {dx}}$

$\therefore$ $v + x{{dv} \over {dx}} =v+ \sqrt {{v^2} + 16} $

$ \Rightarrow $ $ x{{dv} \over {dx}} = \sqrt {{v^2} + 16} $

$ \Rightarrow {{dv} \over {\sqrt {{v^2} + 16} }} = {{dx} \over x}$

Integrating both sides, we get

$\int {{{dv} \over {\sqrt {{v^2} + 16} }} = \int {{{dx} \over x}} } $

$ \Rightarrow \ln \left| {v + \sqrt {{v^2} + 16} } \right| = \ln x + \ln c$

$ \Rightarrow v + \sqrt {{v^2} + 16} = cx$

Now putting, $v = {y \over x}$, we get

${y \over x} + \sqrt {{{{y^2}} \over {{x^2}}} + 16} = cx$

$ \Rightarrow {y \over x} + \sqrt {{{{y^2} + 16{x^2}} \over {{x^2}}}} = cx$

$ \Rightarrow y + \sqrt {{y^2} + 16{x^2}} = c{x^2}$ ...... (1)

Given, $y(1) = 3$

$\therefore$ When x = 1 then y = 3.

Putting in equation (1) we get,

$3 + \sqrt {9 + 16} = c.\,1$

$ \Rightarrow c = 8$

$\therefore$ Solution of equation,

$y + \sqrt {{y^2} + 16{x^2}} = 8{x^2}$

Now, y(2) means when x = 2 then y = ?

$\therefore$ $y + \sqrt {{y^2} + 16 \times 4} = 8 \times 4$

$ \Rightarrow y = 15$

Given differential equation

$2y{e^{{x \over {{y^2}}}}}dx + \left( {{y^2} - 4x{e^{{x \over {{y^2}}}}}} \right)dy = 0,\,x(1) = 0$

$ \Rightarrow {e^{{x \over {{y^2}}}}}[2ydx - 4xdy] = - {y^2}dy$

$ \Rightarrow {e^{{x \over {{y^2}}}}}\left[ {{{2{y^2}dx - 4xydy} \over {{y^4}}}} \right] = {{ - 1} \over y}dy$

$ \Rightarrow 2{e^{{x \over {{y^2}}}}}d\left( {{x \over {{y^2}}}} \right) = - {1 \over y}dy$

$ \Rightarrow 2{e^{{x \over {{y^2}}}}} = - \ln y + c$ ...... (i)

Now, using x(1) = 0, c = 2

So, for x(e), Put y = e in (i)

$2{e^{{x \over {{e^2}}}}} = - 1 + 2$

$ \Rightarrow {x \over {{e^2}}} = \ln \left( {{1 \over 2}} \right) \Rightarrow x(e) = - {e^2}\ln 2$

${{dy} \over {dx}} = 2\tan x(\cos x - y)$

$ \Rightarrow {{dy} \over {dx}} + 2\tan xy = 2\sin x$

$I.F. = {e^{\int {2\tan xdx} }} = {\sec ^2}x$

$\therefore$ Solution of D.E. will be

$y(x){\sec ^2}x = \int {2\sin x{{\sec }^2}xdx} $

$y{\sec ^2}x = 2\sec x + c$

$\because$ Curve passes through $\left( {{\pi \over 4},0} \right)$

$\therefore$ $c = - 2\sqrt 2 $

$\therefore$ $y = 2\cos x - 2\sqrt 2 {\cos ^2}x$

$\therefore$ $\int_0^{\pi /2} {ydx = \int_0^{\pi /2} {(2\cos x - 2\sqrt 2 {{\cos }^2}x)\,dx} } $

$ = 2 - 2\sqrt 2 \,.\,{\pi \over 4} = 2 - {\pi \over {\sqrt 2 }}$

$\left( {{1 \over {\sqrt {1 - {{{y^2}} \over {{x^2}}}} }} + {e^{{y \over x}}}} \right){{dy} \over {dx}} = 1 + \left( {{1 \over {\sqrt {1 - {{{y^2}} \over {{x^2}}}} }} + {e^{{y \over x}}}} \right){y \over x}$

Putting y = tx

$\left( {{1 \over {\sqrt {1 - {t^2}} }} + {e^t}} \right)\left( {t + x{{dt} \over {dx}}} \right) = 1 + \left( {{1 \over {\sqrt {1 - {t^2}} }} + {e^t}} \right)t$

$ \Rightarrow x\left( {{1 \over {\sqrt {1 - {t^2}} }} + {e^t}} \right){{dt} \over {dx}} = 1$

$ \Rightarrow {\sin ^{ - 1}}t + {e^t} = \ln x + C$

$ \Rightarrow {\sin ^{ - 1}}\left( {{y \over x}} \right) + {e^{y/x}} = \ln x + C$

at x = 1, y = 0

So, $0 + {e^0} = 0 + C \Rightarrow C = 1$

at $(2\alpha ,\alpha )$

${\sin ^{ - 1}}\left( {{y \over x}} \right) + {e^{y/x}} = \ln x + 1$

$ \Rightarrow {\pi \over 6} + {e^{{1 \over 2}}} - 1 = \ln (2\alpha )$

$ \Rightarrow \alpha = {1 \over 2}{e^{\left( {{\pi \over 6} + {e^{{1 \over 2}}} - 1} \right)}}$

${{dy} \over {dx}} + {{y(3{x^2} - 1)} \over {x(1 - {x^2})}} = {{4{x^3}} \over {x(1 - {x^2})}}$

$IF = {e^{\int {{{3{x^2} - 1} \over {x - {x^3}}}dx} }} = {e^{ - \ln |{x^3} - x|}} = {e^{ - \ln ({x^3} - x)}} = {1 \over {{x^3} - x}}$

Solution of D.E. can be given by

$y.\,{1 \over {{x^3} - x}} = \int {{{4{x^3}} \over {x(1 - {x^2})}}.\,{1 \over {x({x^2} - 1)}}dx} $

$ \Rightarrow {y \over {{x^3} - x}} = \int {{{ - 4x} \over {{{({x^2} - 1)}^2}}}dx} $

$ \Rightarrow {y \over {{x^3} - x}} = {2 \over {({x^2} - 1)}} + c$

at x = 2, y = $-$2

${{ - 2} \over 6} = {2 \over 3} + c \Rightarrow c = - 1$

at $x = 3 \Rightarrow {y \over {24}} = {2 \over 8} - 1 \Rightarrow y = - 18$

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

${{dx} \over {dy}} + {x \over {1 + {y^2}}} = {{{{\tan }^{ - 1}}y} \over {1 + {y^2}}}$

$I.F. = {e^{\int {{1 \over {1 + {y^2}}}dy} }} = {e^{{{\tan }^{ - 1}}y}}$

$\therefore$ Solution

$x.\,{e^{{{\tan }^{ - 1}}y}} = \int {{{{e^{{{\tan }^{ - 1}}y}}{{\tan }^{ - 1}}y} \over {1 + {y^2}}}dy} $

Let ${e^{{{\tan }^{ - 1}}y}} = t$

${{{e^{{{\tan }^{ - 1}}y}}} \over {1 + {y^2}}} = dt$

$ = x{e^{{{\tan }^{ - 1}}y}} = \int {\ln tdt = t\ln t - t + c} $

$\therefore$ $ = x{e^{{{\tan }^{ - 1}}y}} = {e^{{{\tan }^{ - 1}}y}}{\tan ^{ - 1}}y - {e^{{{\tan }^{ - 1}}y}} + c$ ..... (i)

$\because$ It passes through (1, 0) $\Rightarrow$ c = 2

Now put y = tan1, then

$ex = e - e + 2$

$ \Rightarrow x = {2 \over e}$

${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$

$ = bx\,dy + cy\,dy + a\,dy = ax\,dx - by\,dx + a\,dx$

$ = cy\,dy + a\,dy - ax\,dx - a\,dx + b(x\,dy + y\,dx) = 0$

$ = c\int {y\,dy + a\int {x\,dx - a\int {dx + b\int {d(xy) = 0} } } } $

$ = {{c{y^2}} \over 2} + ay - {{a{x^2}} \over 2} - ax + bxy = k$

$ = a{x^2} - c{y^2} + 2ax - 2ay - 2bxy = k$

Above equation is circle

$\Rightarrow$ a = $-$ c and b = 0

$a{x^2} + a{y^2} + 2ax - 2ay = k$

$ \Rightarrow {x^2} + {y^2} + 2x - 2y = \lambda \,\,\,\,\,\,\,\left[ {\lambda = {k \over a}} \right]$

Passes through (2, 5)

$4 + 25 + 4 - 10 = \lambda \Rightarrow \lambda = 23$

Circle $ \equiv {x^2} + {y^2} + 2x - 2y - 23 = 0$

Centre ($-$1, 1) $r = \sqrt {{{( - 1)}^2} + {1^2} + 23} = 5$

Shortest distance of $(11,6) = \sqrt {{{12}^2} + {5^2}} - 5$

$ = 13 - 5$

$ = 8$

${{dy} \over {dx}} + {{{2^{x-y}}({2^y} - 1)} \over {{2^x} - 1}}=0$, x, y > 0, y(1) = 1

${{dy} \over {dx}} = - {{{2^x}({2^y} - 1)} \over {{2^y}({2^x} - 1)}}$

$\int {{{{2^y}} \over {{2^y} - 1}}dy = - \int {{{{2^x}} \over {{2^x} - 1}}dx} } $

$ = {{{{\log }_e}({2^y} - 1)} \over {{{\log }_e}2}} = - {{{{\log }_e}({2^x} - 1)} \over {{{\log }_e}2}} + {{{{\log }_e}c} \over {{{\log }_e}2}}$

$ = |({2^y} - 1)({2^x} - 1)| = c$

$\because$ $y(1) = 1$

$\therefore$ $c = 1$

$ = |({2^y} - 1)({2^x} - 1)| = 1$

For $x = 2$

$|({2^y} - 1)3| = 1$

${2^y} - 1 = {1 \over 3} \Rightarrow 2y = {4 \over 3}$

Taking log to base 2.

$\therefore$ $y = 2 - {\log _2}3$

$x{{dy} \over {dx}} + 2y = x{e^x},\,\,y(1) = 0$

${{dy} \over {dx}} + {2 \over x}y = {e^x}$, then ${e^{\int {{2 \over x}dx} }}dx = {x^2}$

$y\,.\,{x^2} = \int {{x^2}{e^x}dx} $

$y{x^2} = {x^2}{e^x} - \int {2x{e^x}dx} $

$ = {x^2}{e^x} - 2(x{e^x} - {e^x}) + c$

$y{x^2} = {x^2}{e^x} - 2x{e^x} + 2{e^x} + c$

$y{x^2} = ({x^2} - 2x + 2){e^x} + c$

$0 = e + c \Rightarrow c = - e$

$y(x)\,.\,{x^2} - {e^x} = {(x - 1)^2}{e^x} - e$

$z(x) = {(x - 1)^2}{e^x} - e$

For local maximum $z'(x) = 0$

$\therefore$ $2(x - 1){e^x} + {(x - 1)^2}{e^x} = 0$

$\therefore$ $x = - 1$

And local maximum value $ = z( - 1)$

$ = {4 \over e} - e$

$\because$ ${{dy} \over {dx}} + {e^x}({x^2} - 2)y = ({x^2} - 2x)({x^2} - 2){e^{2x}}$

Here, $I.F. = {e^{\int {{e^x}({x^2} - 2)dx} }}$

$ = {e^{({x^2} - 2x){e^x}}}$

$\therefore$ Solution of the differential equation is

$y\,.\,{e^{({x^2} - 2x){e^x}}} = \int {({x^2} - 2x)({x^2} - 2){e^{2x}}\,.\,{e^{({x^2} - 2x){e^x}}}dx} $

$ = \int {({x^2} - 2x){e^x}\,.\,({x^2} - 2){e^x}\,.\,{e^{({x^2} - 2x){e^x}}}dx} $

Let $({x^2} - 2x){e^x} = t$

$\therefore$ $({x^2} - 2){e^x}dx = dt$

$y\,.\,{e^{({x^2} - 2x){e^x}}} = \int {t\,.\,{e^t}dt} $

$y\,.\,{e^{({x^2} - 2x){e^x}}} = ({x^2} - 2x - 1){e^{({x^2} - 2x){e^x}}} + c$

$\therefore$ $y(0) = 0$

$\therefore$ $c = 1$

$\therefore$ $y = ({x^2} - 2x - 1) + {e^{(2x - {x^2}){e^x}}}$

$\therefore$ $y(2) = - 1 + 1 = 0$

$2{x^2}{{dy} \over {dx}} - 2xy + 3{y^2} = 0$

$ \Rightarrow 2x(xdy - ydx) + 3{y^3}dx = 0$

$ \Rightarrow 2\left( {{{xdy - ydx} \over {{y^2}}}} \right) + 3{{dx} \over x} = 0$

$ \Rightarrow - {{2x} \over y} + 3\ln x = C$

$\because$ $y(e) = {e \over 3} \Rightarrow - 6 + 3 = C \Rightarrow C = - 3$

Now, at $x = 1$, $ - {2 \over y} + 0 = - 3$

$y = {2 \over 3}$

$(x + 1){{dy} \over {dx}} - y = {e^{3x}}{(x + 1)^2}$

${{dy} \over {dx}} - {y \over {x + 1}} = {e^{3x}}(x + 1)$

If ${e^{ - \int {{1 \over {x + 1}}x} }} = {e^{ - \log (x + 1)}} = {1 \over {x + 1}}$

$\therefore$ $y\left( {{1 \over {x + 1}}} \right) = \int {{{{e^{3x}}(x + 1)} \over {x + 1}}dx} $

${y \over {x + 1}} = \int {{e^{3x}}dx} $

${y \over {x + 1}} = {{{e^{3x}}} \over 3} + c$

$\because$ $y(0) = {1 \over 3}$

${1 \over 3} = {1 \over 3} + c$

$\therefore$ $c = 0$

So : $y = {{{e^{3x}}} \over 3}(x + 1)$

$y' = {e^{3x}}(x + 1) + {{{e^{3x}}} \over 3} = {e^{3x}}\left( {x + {4 \over 3}} \right)$

$y'' = 3{e^{3x}}\left( {x + {4 \over 3}} \right) + {e^{3x}} = {e^{3x}}(3x + 5)$

$y' = 0$ at $x = {{ - 4} \over 3}$ & $y'' = {e^{ - 4}}(1) > 0$ at $x = {{ - 4} \over 3}$

$ \Rightarrow x = {{ - 4} \over 3}$ is point of local minima

${{dy} \over {dx}} = {{{y^2}} \over {xy - {x^2} - {y^2}}}$

Put $y = vx$ we get

$v + x{{dv} \over {dx}} = {{{v^2}} \over {v - 1 - {v^2}}}$

$ \Rightarrow x{{dv} \over {dx}} = {{{v^2} - {v^2} + v + {v^3}} \over {v - 1 - {v^2}}}$

$ \Rightarrow \int {{{v - 1 - {v^2}} \over {v(1 + {v^2})}}dv = \int {{{dx} \over x}} } $

${\tan ^{ - 1}}\left( {{y \over x}} \right) - \ln \left( {{y \over x}} \right) = \ln x + c$

As it passes through (1, 1)

$c = {\pi \over 4}$

$ \Rightarrow {\tan ^{ - 1}}\left( {{y \over x}} \right)\ln \left( {{y \over x}} \right) = \ln x + {\pi \over 4}$

Put $y = \sqrt 3 x$ we get

$ \Rightarrow {\pi \over 3} - \ln \sqrt 3 = \ln x + {\pi \over 4}$

$ \Rightarrow \ln x = {\pi \over {12}} - \ln \sqrt 3 = \ln \alpha $

$\therefore$ $\ln \left( {\sqrt 3 \alpha } \right) = \ln \sqrt 3 + \ln \alpha $

$ = \ln \sqrt 3 + {\pi \over {12}} - \ln \sqrt 3 = {\pi \over {12}}$