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KC Sinha Solution Class 12 Chapter 23 अवकल समीकरण (Differential Equations) Exercise 23.4

 Exercise 23.4

Question 1

निम्नलिखित अवकल समीकरण को हल करे।
[Solve the following differential equations]
1.\frac{dy}{dx}=(e^x+1)y
Sol :
\frac{dy}{dx}=(e^x+1)y
\frac{dy}{dx}=(e^x+1)dx

दोनो तरफ समाकलन लेने पर

\int \frac{dy}{dx}=\int (e^x+1)dx

log |y|=ex+x+c


2.\frac{dy}{dx}=\frac{x+1}{2-y},y≠2
Sol :
\frac{dy}{dx}=\frac{x+1}{2-y}

(2-y)dy=(x+1)dx

दोनो तरफ समाकल लेने पर

\int (2-y)dy=\int (x+1)dx

2y-\frac{y^2}{2}=\frac{x^2}{2}+x+k

\frac{4y-y^2}{2}=\frac{x^2+2x+2k}{2}

0=x2+y2+2x-4y+2k

x2+y2+2x-4y+c=0 
जहाँ c=2k


3.(1+x^2)\tan ^{-1} x\frac{dy}{dx}+y=0
Sol :
\frac{dy}{y}=-\frac{dx}{(1+x^2)\tan^{-1}x}

दोनो तरफ समाकल लेने पर

\int \frac{dy}{y}=-\int \frac{dx}{(1+x^2)\tan^{-1}x}

log y=-log |tan-1 x|+log c

log y+log|tan-1 x|=log c

log(y tan-1 x)=log c⇒y tan -1 x=c


4.sec2x tany dx+sec2y tan xdy=0
Sol :
sec2y tan xdy=-sec2x tany dx

\frac{\text{sec}^2 ydy }{\tan y}=\frac{-\text{sec}^2 x}{\tan x}dx

Integrating both side

\int \frac{\sec^2 y}{\tan y}dy=-\int \frac{\sec^2 x}{\tan x}dx

log |tan y|=-log |tan x|+log c

log |tan x|+\log |\tan y|=\log c

log|tan x. tan y)|=log c

tan x tan y=c


5.ydx+xdy=xy(dy-dx)

Sol :

ydx+xdy=xydy-xydx

xdy-xydy=-ydx-xydx

x(1-y)dy=-y(1+x)dx

\frac{(1-y)}{y}dy=-\frac{(1+x)}{x}dx

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

Integrating both side

\int \left(\frac{1}{y}-1\right)dy=-\int \left(\frac{1}{x}+1\right)dx

log y-y=-(log x+x)+c

log x+x+log y-y=c

log xy +x-y=c


6.(1-x2)dy+xydy=xy2dx

Sol :

(1+x2)dy=xy2dx-xydx

(1-x2)dy=xy(y-1)dx

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

Integrating both side

\int \frac{dy}{y(y-1)}=\int \frac{x}{1-x^2}dx

\int \left[\frac{1}{y-1}-\frac{1}{y}\right]dy=-\frac{1}{2}\int \frac{-2x}{1-x^2}dx

log |y-1|-log|y|=-\frac{1}{2}\log|1-x^2|+\log c

\log \frac{y-1}{y}+\log (1-x^2)^{\frac{1}{2}}=\log c

\log \frac{y-1}{y}\sqrt{1-x^2}=\log c

\frac{y-1}{y}\sqrt{1-x^2}=c


7.tan ydx+secy tan xdy=0
Sol :
secy tanxdy=-tan ydx

\frac{\sec ^2 y}{\tan y}dy=\frac{-dx}{\tan x}

Integrating both side

\int \frac{\text{sec }^2y}{\tan y}dy=-\int \text{cot x}dx

log|tan y|=-log|sin x|+log c

log |sin x|+log|tan y|=log c

log|sin x.tan y|=log c

sin x.tan y=c


8.\text{cosec x}log y \frac{dy}{dx}+x^2y=0
Sol :
\text{cosec x}\log y \frac{dy}{dx}=-x^2y

\frac{\log y}{y}dy=\frac{-x^2 dx}{\text{cosec x}}

Integrating both sides

\int \frac{\log y}{y}dy=-\int x^2\sin x dx


log y=t

Differentiating w.r.t y

\frac{1}{y}=\frac{dt}{dy} 

\frac{dy}{y}=dt


\int t dt=-2\left[x^2 \int \sin x dx -\int \left\{\frac{d(x^2)}{dx}.\int \sin x dx \right\}dx\right]

\frac{1}{2}t^2=-\left[x^2 \cos x-\int 2x \cos x dx \right]

\frac{1}{2} (\log y)^2 =+x^2 \cos x-2 \int x \cos x dx

\frac{1}{2}(\log y)^2=x^2 \cos x-2 \left[x\int \cos xdx-\int \left\{\frac{dx}{dx}\int \cos x dx\right\}dx\right]

\frac{1}{2}(\log y)^2=x^2 \cos x -2\left[x \sin x-\int \sin x dx\right]

\frac{1}{2}(\log y)^2=x^2 \cos x-2x \sin x+2(-\cos x)+c

\frac{1}{2}(\log y)^2-x^2\cos x+2x\sin x+2\cos x=c


9.(x2-yx2)dy+(y2+x2y2)dx=0
Sol :
(x2-yx2)dy=-(y2+x2y2)dx
x2(1-y)dy=-y2(1+x2)dx
\frac{(1-y)dy}{y^2}=-\frac{(1+x^2)}{x^2}dx
\left(\frac{1}{y^2}-\frac{1}{y}\right)dy=-\left(\frac{1}{x^2}+1\right)dx

Integrating both side

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

-\frac{1}{y}-\log |y|=-\left(-\frac{1}{x}+x\right)+c

x-\frac{1}{x}-\frac{1}{y}-\log |y|=c


10.\frac{dy}{dx}=(1+x^2)(1+y^2)
Sol :
\frac{dy}{1+y^2}=(1+x^2)dx

Integrating both sides

\int \frac{dy}{1+y^2}=\int (1+x^2)dx

\tan ^{-1} y=x+\frac{x^3}{3}+c



11. 2sin3ydx+3xcos3ydy=0
Sol :
2sin3ydx+3xcos3ydy=0

3xcos3ydy=-2sin3ydy

Integrating both sides

\int \frac{\cos 3y}{\sin 3y}dy=-\frac{2}{3}\int \frac{1}{x}dx

\frac{\log|\sin 3y|}{3}=\frac{-2}{3}\log x+\log k

\frac{\log|\sin 3y|}{3}=\frac{-\log x^2 +3 \log k}{3}

log|sin 3y|+log x2=3log k

log|xsin 3y|=log k3

xsin 3y=k3

xsin 3y=c ,जहाँ c=k3

12.\frac{dy}{dx}=\frac{x}{x^2-1}
Sol :
dy=\frac{x}{x^2+1}dx

Integrating both sides

\int dy=\frac{1}{2}\int \frac{2x}{x^2+1}dx

y=\frac{1}{2}\log|x^2+1|+c


13.e^x+e^{-x}\frac{dy}{dx}=(e^x-e^{-x})
Sol :
dy=\frac{(e^x-e^{-x})}{e^x+e^{-x}}dx

Integrating both side

\int dy=\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}

y=log|ex+e-x|+c


14.(x+2)\frac{dy}{dx}=x^2+4x-9
Sol :
dy=\frac{x^2+4x-9}{x+2}dx

dy=\frac{(x^2+4x+4)-13}{(x+2)}dx

dy=\frac{(x+2)^2-13}{x+2}dx

dy=\left[x+2-\frac{13}{x+2}\right]

Integrating both sides

\int dy=\int \left(x+2-\frac{13}{x+2}\right)

y=\frac{x^2}{2}+2x-13\log |x+2|+c

15.\frac{dy}{dx}=\sin ^3 x\cos ^2 x+xe^x
Sol :
dy=(sin3 x cosx+xex)dx

dy=[sin x(sin2 x)cos2 x+xex]dx$

dy=[sin x(1-cos2x cos2 x+xex]dx$

dy=[sin x cosx-sin cosx+xex]dx

Integrating both sides

\int dy=\int (\sin x \cos^2 x-\sin x \cos^4 x+xe^x)dx


Putting cosx=t

Differentiating w.r.t x

-\sin x=\frac{dt}{dx}

sin xdx=-dt


y=-\frac{1}{3}\cos ^3 x+\frac{1}{5}\cos ^5 x+x\int e^x dx-\int \left\{\frac{d(x)}{dx}\int e^x dx\right\}dx

y=-\frac{1}{3}\cos ^3 x+\frac{1}{5} \cos ^5 x+x e^x -\int e^x dx

y=-\frac{1}{3}\cos ^3 x+\frac{1}{5}\cos ^5 x+ xe^x-e^x +c


16.\frac{dy}{dx}=\frac{1-\cos x}{1+\cos x}dx
Sol :
dy=\frac{1-\cos x}{1+\cos x}dx

dy=\frac{2\sin ^2 \frac{x}{2}}{2 \cos ^2 \frac{x}{2}}dx

dy=\tan^{2} \frac{x}{2} dx

dy=\left(\sec ^2 \frac{x}{2}-1\right)dx

Integrating both sides

\int dy=\int \left(\sec ^2 \frac{x}{2}-1\right)dx

y=2\tan \frac{x}{2}-x+c


17.ylog ydx-xdy=0
Sol :
-xdy=-y log ydx

\frac{dy}{y log y}=\frac{dx}{x}

Integrating both sides

\int \frac{dy}{y log y}=\int \frac{dx}{x}

log |log y|=log |x|+log c

log |log y|=log |cx|

log y= cx

y=ecx


18.\frac{dy}{dx}+y=1,y≠1
Sol :
\frac{dy}{dx}+y=1

\frac{dy}{dx}=1-y

dy=(1-y)dx

\frac{dy}{1-y}=dx

Integrating both sides

\int \frac{dy}{1-y}=\int dx

-log|1-y|=x+c

0=x+log|1-y|+c

19.(x+1)\frac{dy}{dx}=2xy
Sol :
\frac{dy}{dx}=\frac{2x}{x+1}dx

Integrating both sides

\int \frac{dy}{dx}=2\int \frac{x}{x+1}dx

\int \frac{dy}{dx}=2\int \left[1-\frac{1}{x+1}\right]dx

log |y|=2[x-log|x+1|]+c

20.x^5\frac{dy}{dx}=-y^5
Sol :
\frac{dy}{y^5}=-\frac{dx}{x^5}

y-5dy=-x-5dx

Integrating both sides

\int y^{-5}dy=-\int x^{-5}dx

\frac{y^{-4}}{-4}=-\frac{x^{-4}}{-4}+k

-\frac{x^{-4}}{4}-\frac{y^{-4}}{4}=k

-\frac{1}{4x^4}-\frac{1}{4y^4}=k

\frac{-(y^4+x^4)}{4x^4y^4}=k

x4+y4=-4kx4y4

x4+y4=cx4+y4

जहाँ c=-4k


21.e^x\sqrt{1-y^2}dx+\frac{y}{x}dy=0
Sol :
\frac{y}{x}dy=e^x\sqrt{1-y^2}dx

\frac{y}{\sqrt{1-y^2}}dy=xe^x dx

Integrating both sides

-\frac{1}{2}\int \frac{-2y}{\sqrt{1-y^2}}dy=\int xe^x dx

-\frac{1}{2}\times 2 \sqrt{1-y^2}=x\int e^x dx-\int \left\{\frac{dx}{dx}\int e^x dx\right\}dx

-\sqrt{1-y^2}+c=xe^x-e^x


22.x(1+y2)dx-y(1-x2)dy=0
Sol :
-y(1-x2)dy=-x(1+y2)dy=0

\frac{y}{1+y^2}dy=\frac{x}{1-x^2}dx

Integrating both sides 

\frac{1}{2}\int \frac{2y}{1+y^2}dy=\frac{1}{2}\int \frac{-2x}{1-x^2}dx

\frac{1}{2}\log |1+y^2|=-\frac{1}{2}\log|1-x^2|+\log c

\frac{1}{2}\log(1-x^2)(1+y^2)=\log c

log(1-x2)(1+y2)=2log c

log(1-x2)(1+y2)=log c2

(1-x2)(1+y2)=c2

(1-x2)(1+y2)=k , जहाँ k=c2


23.\frac{dy}{dx}=1+x+y+xy
Sol :
\frac{dy}{dx}=1(1+x)+y(1+x)

\frac{dy}{dx}=(1+x)(1+y)

\frac{dy}{1+y}=(1+x)dx

दोनो तरफ समाकलन लेने पर

\int \frac{dy}{1+y}=\int (1+x)dx

log |1+y|=x+\frac{x^2}{2}+c


24.etan ydx+(1-ex)secydy=0
Sol :
(1-ex)secydy=-etan ydx

\frac{\sec ^2 y}{\tan y}dy=\frac{-e^x}{1-e^x}dx

Integrating both sides

\int \frac{\sec ^2 y}{\tan y}dy=\int \frac{-e^x}{1-e^x}dx

log|tan y|=log|1-ex|+log |c|

log|tan y|=log|(1-ex)c|

tan y=(1-ex)c


25.\frac{dy}{dx}=e^{x-y}+x^2 e^{-7}
Sol :
\frac{dy}{dx}=e^{-y}(e^x+x^2)

\frac{dy}{e^{-y}}=(e^x+x^2)dx

ey dy=(ex+x2)dx

Integrating both sides

\int e^y dy=\int (e^x+x^2)dx

e^y=e^x+\frac{x^3}{3}+c


26.\log_e \left(\frac{dy}{dx}\right)=3x-5y
Sol :
\frac{dy}{dx}=e^{3x-5y}

\frac{dy}{dx}=\frac{e^{3x}}{e^{5y}}

e5y dy=e3x dx

Integrating both sides

\int e^{5y}dy=\int e^{3x}dx

\frac{e^{5y}}{5}=\frac{e^{3x}}{3}+k

\frac{e^{5y}}{5}-\frac{e^{3x}}{3}=k

\frac{3e^{5y}-5e^{3x}}{15}=k

3e5y -5e3x=15k

3e5y -5e3x=15k , जहाँ c=15k


27.\frac{dy}{dx}=e^{x+y}
Sol :
\frac{dy}{dx}=e^x.e^y

\frac{dy}{dx}=e^x dx

e-ydy=edx

Integrating both sides

\int e^{-y}dy=\int e^{x}dx

-e-ydy=ex+c


28.x(xdy-ydx)=ydx
Sol :
x2dy-xydx=y dx

x2dy=ydx+xydx

x2dy=y(1+x)dx

\frac{dy}{y}=\frac{(1+x)}{x^2}dx

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

log |y|=-\frac{1}{x}+\log|x|+\log|c|

log|y|=-\frac{1}{x}+\log|cx|

log|y|-log|cx|=-\frac{1}{x}

\log \left|\frac{y}{cx}\right|=-\frac{1}{x}

\frac{y}{cx}=e^{-\frac{1}{x}}

y=cxe^{-\frac{1}{x}}


29.प्रारम्भिक मान प्रश्न [Solve the initial value problem] y'=y\cot 2x , y\left(\frac{\pi}{4}\right)=2 को हल करे।
Sol :
y'=ycot2x

\frac{dy}{dx}=y\cot 2x

\frac{dy}{y}=\cot 2x dx

Integrating both sides

\int \frac{dy}{dx}=\int \cot 2x dx

log|y|=\frac{1}{2}\log |\sin 2x|+\log |c|

2log|y|=log|c sin 2x|

log|y2|=log|c sin 2x|

y2=c sin 2x

y\left(\frac{\pi}{4}\right)=2

2^2=c \sin 2\left(\frac{\pi}{4}\right)

4=c

∴Solution of differentiating equation

y2=4 sin 2x

30.अवकल समीकरण [Solve the differential equation] (x-1)\frac{dy}{dx}=2xy को हल करे ।
Sol :
(x-1)\frac{dy}{dx}=2xy

\frac{dy}{y}=\frac{2x}{x-1}dx

Integrating both sides

\int \frac{dy}{y}=\int \frac{2x}{x-1}dx

\int \frac{dy}{y}=\int \left(2+\frac{2}{x-1}\right)dx

log|y|=2x+2log|x-1|+c


31.अवकल समीकरण [Solve the differential equation] (1+x2)dy=xydy को हल करे।
Sol :
(1+x2)dy=xydy

\frac{dy}{y}=\frac{x}{1+x^2}dx

Integrating both sides

\int \frac{dy}{y}=\frac{1}{2}\int \frac{2x}{1+x^2}dx

log|y|=\frac{1}{2}\log |1+x^2|+\log k

log |y2|=log|1+x2|+log k

log |y2|=log|k(1+x2)|

y2=k(1+x2)

y=\sqrt{k}\sqrt{1+x^2}

y=c\sqrt{1+x^2}
जहाँ c=√k


32.अवकल समीकरण [Solve the differential equation] xy(y+1)dy=(x2+1)dx को हल करे।
Sol :
xy(y+1)dy=(x2+1)dx

y(y+1)dy=\frac{(x^2+1)}{x}dx

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

Integrating both sides

\int (y^2+y)dy=\int (x+\frac{1}{x})dx

\frac{y^3}{3}+\frac{y^2}{2}=\frac{x^2}{2}+\log|x|+c


33.अवकल समीकरण [Solve the differential equation] 5\frac{dy}{dx}=e^{x}y^4 को हल करे।
Sol :
5\frac{dy}{dx}=e^{x}y^4

\frac{5dy}{y^4}=e^xdx

5y-4 dy=ex dx

Integrating both sides

5\int y^{-4}dy=\int e^x dx

\frac{5y^{-3}}{-3}=e^x+c

\frac{-5}{3y^3}=e^x+c

34.अवकल समीकरण  को हल करे। [Solve the differential equation]

sinxdx-sin ydy=0 

Sol :

-sin ydy=-sinxdx

Integrating both sides

\int \sin ydy=\frac{1}{4}\int 4\sin ^3 xdx

\int \sin ydy=\frac{1}{4}\int (3\sin x-\sin 3x)dx

-\cos y+c=\frac{1}{4} \left[-3 \cos x+\frac{\cos 3x}{3}\right]dx

-\cos y+c=\frac{-3}{4}\cos x+\frac{1}{12}\cos 3x

c=\cos y-\frac{3}{4}\cos x+\frac{1}{12}\cos 3x


35.निम्नलिखित अवकल समीकरणो को हल करे।

[Solve the following differential equations]:

x\frac{dy}{dx}+y=y^2

Sol :
x\frac{dy}{dx}=y^2-y

\frac{dy}{y^2-y}=\frac{dx}{x}

\frac{dy}{y(y-1)}=\frac{dx}{x}

Integrating both sides

\int \frac{dy}{y(y-1)}=\int \frac{dy}{x}

\int \left[\frac{1}{y-1}-\frac{1}{y}\right]dy=\int \frac{dx}{x}

log|y-1|-log|y|=log|x|+log c

log|y-1|=log|cxy|

y-1=cxy


36.\frac{dy}{dx}=\frac{3e^{2x}+3e^{4x}}{e^x+e^{-x}}

Sol :

dy=\frac{3e^{2x}(1+e^{2x})}{e^x+\frac{1}{e^x}}dx

dy=\dfrac{3e^{2x}(1+e^{2x})}{\frac{e^{2x}+1}{e^x}}dx

dy=3e3x dx

Integrating both sides

\int dy=3\int e^{3x}dx

y=3\frac{e^{3x}}{3}+c

y=e3x +c


37.\sqrt{1+x^2}dy+\sqrt{1+y^2}dx=0

Sol :

\sqrt{1+x^2}dy=-\sqrt{1+y^2}dx=0

\frac{dy}{\sqrt{1-y^2}}=\frac{-dx}{\sqrt{1+x^2}}

Integrating both sides

\int \frac{dy}{\sqrt{1^2+y^2}}=-\int \frac{dx}{\sqrt{1^2+x^2}}

\log|y+\sqrt{1+y^2}|=-\log|x+\sqrt{1+x^2}|+\log c

log|(y+√1+y2).(x+√1+x2)|=log c

(y+√1+y2)(x+√1+x2)=c


38.\tan y\frac{dy}{dx}=\sin (x+y)+\sin (x-y)
Sol :
\tan y\frac{dy}{dx}=\sin x \cos y+\cos x \sin y+\sin x \cos y-\cos x \sin y

\tan y\frac{dy}{dx}=2\sin xdx

sec y tan y dy=2 sin xdx

Integrating both sides

\int \sec y \tan y dy=2\int \sin x dx

sec y=-2cos x+c


39.अवकल समीकरण \frac{dy}{dx}=-4xy^2 का विशिष्ट हल ज्ञात कीजिए यदि दिया है कि y=1 जब x=0
Sol :
\frac{dy}{dx}=-4xy^2

\frac{dy}{y^2}=-4xdx

Integrating both sides

\int \frac{dy}{y^2}=-4 \int xdx

-\frac{1}{y}=-4\frac{x^2}{2}+c

2x^2-\frac{1}{y}=c

y=1 जब x=0

2(0)^2-\frac{1}{1}=c

-1=c


∴विशिष्ट हल

2x^2-\frac{1}{y}=-1

2x^2+1=\frac{1}{y}

\frac{1}{2x^2+1}=y


40.अवकल समीकरण \frac{dy}{dx}=y \tan x का विशिष्ट हल ज्ञात कीजिए यदि दिया है कि y=1 जब x=0

Sol :

\frac{dy}{dx}=y \tan x

\frac{dy}{y}=\tan x dx

Integrating both sides

\int \frac{dy}{y}=\int \tan x dx

log|y|=log|sec x|+log c

log|y|=log |c sec x|

y=c sec x

y=1 , जब x=0

1=c sec 0

1=c

∴अवकल समीकरण का विशिष्ट हल 

y=sec x


41.अवकल समीकरण x(x^2-1)\frac{dy}{dx}=1 का विशिष्ट हल ज्ञात कीजिए यदि दिया है कि x=2 जब y=0

Sol :

x(x^2-1)\frac{dy}{dx}=1

dy=\frac{dx}{x(x^2-1)}

dy=\left[\frac{x}{x^2-1}-\frac{1}{x}\right]dx

Integrating both sides

\int dy=\int \left(\frac{x}{x^2-1}-\frac{1}{x}\right)dx

\int dy=\frac{1}{2}\int \frac{2x}{x^2-1}dx-\int \frac{1}{x}dx

y=\frac{1}{2}\log|x^2-1|-\log|x|+c

x=2 , y=0 पर

0=\frac{1}{2}\log|2^2-1|-\log|2|+c

0=\frac{1}{2}\log 3-log 2+c

c=-\frac{1}{2}\log 3+\log 2


∴अवकल समीकरण का हल

y=\frac{1}{2}\log|x^2-1|-\log|x|+\log 2-\frac{1}{2}\log 3


42.अवकल समीकरण cos ydy+cos x sinydx=0 का विशिष्ट हल ज्ञात कीजिए यदि दिया है कि y=\frac{\pi}{2} जब x=\frac{\pi}{2}

Sol :

cos ydy+cos x sin ydx=0

cos ydy=-cos x sin ydx

\frac{\cos y}{\sin y}dy=-\cos xdx

Integrating both sides

\int \frac{\cos y}{\sin y}dy=-\int cos xdx

log|sin y|=-sin x+c

log|sin  y|+sin x=c


At x=\frac{\pi}{2} , y=\frac{\pi}{2}

\log|\sin \frac{\pi}{2}|+\sin \frac{\pi}{2}=c

log 1+1=c

1=c

∴अवकल समीकरण का हल

log|sin y|+sin x=1


43.अवकल समीकरण (x-1)\frac{dy}{dx}=2xy को हल करे यदि y=1 जब x=2 

Sol :

(x-1)\frac{dy}{dx}=2xy

\frac{dy}{y}=\frac{2x}{x-1}dx

Integrating both sides

\int \frac{dy}{y}=2\int \frac{x}{x-1}dx

\int \frac{dy}{y}=2 \int \left[1+\frac{1}{x-1}\right]dx

log y=2[x+log(x-1)]+c

log y=2x+2log(x-1)+c

log y-log(x-1)2=2x+c

\log \frac{y}{(x-1)^2}=2x+c


y=1 जब x=2

\log \frac{1}{(2-1)^2}=2(2)+c

\log \frac{1}{1}=4+c

c=-4

∴अलकल समीकरण का हल:

\log \frac{y}{(x-1)^2}=2x-4

\log \frac{y}{(x-1)^2}=2(x-2)

\frac{y}{(x-1)^2}=e^{2(x-2)}

y=(x-1)2e2(x-2)


44..अवकल समीकरण [Solve the differential equation] (1+y)2dx+xdy=0 यदि y(1)=1 [given that y(1)=1]

Sol :

(1+y)2dx+xdy=0

xdy=-(1+y2)dx

\frac{dy}{1+y^2}=-\frac{1}{x}dx

Integrating both sides

\int \frac{dy}{1+y^2}=-\int \frac{1}{x}dx

tan-1 y-log x+c

log x+tan-1 y=c


y=1, जब x=1

log 1+tan-1 1=c

\frac{\pi}{4}=c

∴अवकल समीकरण का हल 

\log x+\tan ^{-1}y=\frac{\pi}{4}


45.\cos \frac{dy}{dx}=a का विशिष्ट हल ज्ञात करे यदि y=2 जब x=0

Sol :

\cos \frac{dy}{dx}=a

\frac{dy}{dx}=\cos ^{-1}a

dy=cos-1a

Integrating both sides

\int dy=\cos ^{-1}a \int dx

y=x cos-1a+c

y=2 , जब x=0

2=(0)cos-1a+c

2=c

∴अवकल समीकरण का विशिष्ट हल:

y=xcos-1a+2


46.उस वक्र का समीकरण ज्ञात करे जो बिन्दु(1,1) से गुजरे तथा अवकल समीकरण xdy=(2x2+1)dx ,x≠0 है

Sol :

xdy=(2x2+1)dx

dy=\frac{(2x^2+1)}{x}dx

dy=\left(2x+\frac{1}{x}\right)dx

Integrating both sides

\int dy=\int \left(2x+\frac{1}{x}\right)dx

y=2\frac{x^2}{2}+\log|x|+c

y=x2+log|x|+c

x=1 ,y=1

1=12+log|1|+c

1=1+0+c

c=0

∴वक्र का समीकरण

y=x2+log|x|


47.उस वक्र का समीकरण ज्ञात करे जो बिन्दु(0,0) से गुजरे तथा अवकल समीकरण y'=esin x है

Sol :

y'=esin x

\frac{dy}{dx}=e^x \sin x

dy=esin x dx

Integrating both sides

\int dy=\int e^x \sin x dx

y=\int e^x \sin x dx

Let I=\int e^x \sin x dx

I=\sin x \int e^x dx-\int \left\{\frac{d(\sin x)}{dx}\int e^x dx\right\}dx

I=e^x \sin x-\int \cos x e^x dx

I=e^x \sin x-\left[\cos x \int e^x dx-\int \left\{\frac{d(\cos x)}{dx}\int e^x dx\right\}dx\right]

I=e^x \sin x-\left[e^x \cos x-\int \sin x e^x dx\right]

I=e^x \sin x-e^x \cos x-I

2I=ex(sin x-cos x)

I=\frac{1}{2}e^x \left(\sin x-\cos x\right)

y=\int e^x \sin xdx

y=\frac{1}{2}e^x (\sin x-\cos x)+c

y=\frac{1}{2}e^x (\sin x-\cos x)+c


At x=0, y=0

0=\frac{1}{2}e^0(\sin 0-\cos 0)

0=\frac{1}{2}(0-1)+c

\frac{1}{2}=c

वक्र का समीकरण 

y=\frac{1}{2}e^x(\sin x-\cos x)+\frac{1}{2}

2y=e^x(\sin x-\cos x)


48.एक गोलाकार गुब्बारे का आयतन, जिसे हवा भरकर फुलाया जा रहा है, स्थिर गति से बदल रहा है यदि आरंभ मे इस गुब्बारे की त्रिज्या 3 इकाई है और 3 सेकेंड बाद 6 इकाई है, तो t सेकेंड बाद उस गुब्बारे की त्रिज्या ज्ञात कीजिए।

Sol :

माना गोलाकार गुब्बारे की त्रिजया r तथा आयतन=v

आयतन मे परिवर्तन की दर, \frac{dv}{dt}=k (माना)

\dfrac{d\left(\frac{4}{3}\pi r^3\right)}{dt}=k

4\pi r^2 \frac{dr}{dt}=k

4πrdr=k dt

Integrating both sides

4\pi \int r^2 dr=k\int dt

\frac{4}{3}\pi r^3=kt+c

जब, r=3, t=0

\frac{4}{3}\pi (3)^3

=k(0)+c

36π=c

जब, r=6, t=3

\frac{4}{3}\pi(1)^3=k(3)+36\pi

288π=3k+36π

252π=3k

k=\frac{252 \pi}{3}

k=84π


c=36π ,k=84k , t समय मे त्रिज्या

\frac{4}{3}\pi r^3=84 \pi t+36 \pi

r=63t+27

r=(63t+27)^{\frac{1}{3}}


49.किसी बैक मे मूलधन की वृद्धि r% वार्षिक की दर से होती है। यदि 100 रूपये 10 वर्षो मे दुगुने हो जाते है, तो r का मान ज्ञात कीजिए । (loge=20.6931)

Sol :

माना t समय मे मूलधन P है।

\frac{dP}{dt}=P का r%

\frac{dP}{dt}=P\times \frac{r}{100}

\frac{dP}{dt}=\frac{r}{100}dt

Integrating both sides

\int \frac{dP}{p}=\frac{r}{100}\int dt

log|P|=\frac{r}{100}t+C


जब ,t=0, P=100

log|100|=\frac{r}{100}\times 0+C

c=log 100


जब ,t=10 वर्ष , P=200

log 200=\frac{r}{100}(10)+log 100

log 200-log 100=\frac{r}{10}

\log \left|\frac{200}{100}\right|=\frac{r}{10}

0.6931=\frac{r}{10}

r=6.931%

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