Differential Equations Test

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Differential Equations Test - 21

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Weekly Quiz Competition

Question 1
1 / -0

The solution of $$\dfrac{dy}{dx}=\dfrac{x^{2}+4x-9}{x+2}$$ is:

A
$$y=(x+2)^{2}-13log\left | x+2 \right |+c$$

B
$$y=(x+2)^{2}-5log\left | x+2 \right |+c$$

C
$$y=\frac{x^{2}}{2}+2x+13log\left | x+2 \right |+c$$

D
$$y=\frac{x^{2}}{2}+2x-13log\left | x+2 \right |+c$$

Solution

$$\dfrac{dy}{dx}=\dfrac{(x+2)^{2}-13}{x+2}$$
$$\dfrac{dy}{dx}=(x+2)-\dfrac{13}{x+2}$$
$$\int dy=\int (x+2)dx-\int \dfrac{13}{x+2}dx$$
$$\Rightarrow y=\dfrac{x^{2}+4x}{2}-13\ log(x+2)+c$$
Question 2
1 / -0

Solution of $$(e^{x}+1)ydy+(y+1)dx=0$$ is:

A
$$(y+1)(1+e^{-x})=ce^{y}$$

B
$$(e^{x}+1)y=c$$

C
$$(1+e^{x})(y+1)=c$$

D
$$(e^{x}+1)x=c$$

Solution

$$(e^{x}+1)ydy+(y+1)dx=0$$
$$\Rightarrow \left ( \dfrac{y}{y+1} \right )dy+\left ( \dfrac{e^{-x}}{1+e^{-x}} \right )dx=0$$
$$\Rightarrow \int \left ( 1-\dfrac{1}{y+1} \right )dy-\int \left ( \dfrac{-e^{-x}}{1+e^{-x}} \right )dx=-c$$
$$\Rightarrow y-log(y+1)-log (1+e^{-x})=-c$$
$$y+c=log(y+1)(1+e^{-x})$$
$$\Rightarrow (y+1)(1+e^{-x})=c\cdot e\ ^{y}$$
Question 3
1 / -0

The solution of $$(x^{2}-y^{2}x^{2})\dfrac{dy}{dx}+(y^{2}+x^{2}y^{2})=0$$ is:

A
$$x+y+\dfrac{1}{x}+\dfrac{1}{y}=c$$

B
$$2(x+y)-\left ( \dfrac{1}{x}+\dfrac{1}{y} \right )=c$$

C
$$2(x-y)+\left ( \dfrac{1}{x}-\dfrac{1}{y} \right )=c$$

D
$$(x-y)-\left ( \dfrac{1}{x}+\dfrac{1}{y} \right )=c$$

Solution

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

$$\displaystyle \Rightarrow \int { \dfrac { dy \left( { y }^{ 2 }-1 \right) }{ { y }^{ 2 } } dx } =\int { \left( \dfrac { 1 }{ { x }^{ 2 } } +1 \right) } dx$$

$$\displaystyle \Rightarrow \dfrac { 1 }{ y } +y=-\dfrac { 1 }{ x } +x+c\Rightarrow \left( x-y \right) -\left( \dfrac { 1 }{ x } +\dfrac { 1 }{ y } \right) =c$$
Question 4
1 / -0

The solution of $$\frac{dy}{dx}=e^{3x-2y}+x^{2}e^{-2y}$$

A
$$y=log(e^{3x}+x^{2})+c$$

B
$$y=2log(e^{3x}+x^{2})+c$$

C
$$2e^{2x}=3(e^{3y}+y^{3})+c$$

D
$$3e^{2y}=2(e^{3x}+x^{3})+c$$

Solution

$$\dfrac{dy}{dx}=e^{-2y}\left ( e^{3x}+x^{2} \right )$$
$$\Rightarrow \int e^{2y}dy=\int \left ( e^{3x}+x^{2} \right )dx+c$$
$$\Rightarrow \dfrac{e^{2}}{2}=\dfrac{e^{3x}}{3}+\ ^{x\ 3}/_{3}+c$$
$$3e^{2y}=2\left ( e^{3x}+x^{3} \right )+c$$
Question 5
1 / -0

The solution of $$\frac{dy}{dx}=ysin(2x)$$ given that $$y(0)=0$$

A
$$log\left | y \right |=sin^{2}x$$

B
$$log\left | y \right |=sinx$$

C
$$log\left | y \right |=sin2x$$

D
$$log\left | y \right |=cos^{2}x$$

Solution

$$\dfrac{dy}{dx}=y sin2x$$

$$\Rightarrow \int \dfrac{dy}{y}=\int sin2xdx-logC$$

$$\Rightarrow log|y|=\dfrac{-cos2x}{2}-log C$$

$$\Rightarrow log|y|=sin^{2}x-log C$$
Question 6
1 / -0

The solution of $$x\dfrac{dy}{dx}=2\sqrt{y-1}$$ is:

A
$$e^{\sqrt{y-1}}=cx$$

B
$$\sqrt{y-x}=ex$$

C
$$\dfrac{1}{\sqrt{y-1}}=cx$$

D
$$x(\sqrt{y-x})=c$$

Solution

$$\Rightarrow (y-1)^{^{-1}/_{2}}dy=\dfrac{2dx}{x}$$
$$\Rightarrow \int (y-1)^{^{-1}/_{2}}dy=\dfrac{2dx}{x}$$
$$\Rightarrow \dfrac{(y-1)^{1}/_{2}}{^{1}/_{2}}=2log\ x+c$$
$$\Rightarrow 2(y-1)^{^{1}/_{2}}=2log\ x+log\ c$$
$$\Rightarrow (y-1)^{^{1}/_{2}}=log\ x\ c$$
$$\Rightarrow e^{\sqrt{y-1}}=cx$$
Question 7
1 / -0

The solution of $$ydx+xdy=dx+dy$$ is:

A
$$xy=x+y+c$$

B
$$x-y\dfrac{x}{y}+c=0$$

C
$$xy-x+y=c$$

D
$$x+y\dfrac{x}{y}+c=0$$

Solution

$$\Rightarrow ydx+xdy=dx+dy$$
$$\Rightarrow (y-1)dx+(x-1)dy=0$$
$$\Rightarrow \dfrac{dy}{(y-1)}=\dfrac{-dx}{x-1}$$
$$\Rightarrow log(y-1)=-log(x-1)+loge$$
$$\Rightarrow log(x-1)(y-1)=log c$$
$$\Rightarrow (x-1)(y-1)=c$$
$$xy=x+y+c$$
Question 8
1 / -0

The solution of $$x^{2}\dfrac{dy}{dx}=\sqrt{4-y^{2}}$$ is:

A
$$sin^{-1}\left ( \dfrac{y}{2} \right )+\dfrac{1}{x}=c$$

B
$$\dfrac{x^{3}}{3}+sin^{-1}\left ( \dfrac{y}{4} \right )=c$$

C
$$\sin^{-1}\left ( \dfrac{y}{2} \right )+2x=c$$

D
$$\sqrt{4-y^{2}}+2x=c$$

Solution

$$\Rightarrow \dfrac{dy}{\sqrt{4-y^{2}}}=\dfrac{dx}{x^{2}}$$
$$\displaystyle\Rightarrow \int \dfrac{dy}{\sqrt{4-y^{2}}}=\int \dfrac{dx}{x^{2}}+c$$
$$\Rightarrow sin^{-1}({y/{2}})=\ {-1}/{x}+c$$
$$\Rightarrow sin^{-1}(\dfrac{y}{2})+{1}/{x}=c$$
Question 9
1 / -0

The solution of $$xcos^{2}ydx=ycos^{2}xdy$$ is:

A
$$\tan x \tan y =c$$

B
$$y \tan y =x \tan x + c$$

C
$$\tan x \cos y = \tan y \cos x +c$$

D
$$y \tan y - x \tan x + log \left( \dfrac{cosy}{cosx} \right )=c$$

Solution

$$\Rightarrow \int x\ sec^{2}x\ dx=\int y\ sec^{2}ydy+c$$
$$\Rightarrow x\ tan\ x-\int tan\ x\ dx=y\ tan\ y-\int tan\ y+c$$
$$\Rightarrow x\ tan\ x-log\ sec\ x=y\ tan\ y-log\ sec\ y+c$$
$$\Rightarrow y\ tan\ y-x\ tan\ x+log\frac{cos\ y}{cos\ x}=c$$
Question 10
1 / -0

The solution of $$y (dx) +(1+x^{2})tan^{-1} x (dy)=0$$

A
$$ytan^{-1}x=c$$

B
$$xtan^{-1}x=0$$

C
$$y(1+x^{2})=c$$

D
$$y^{2}(1+x^{2})=c$$

Solution

$$\Rightarrow \dfrac{dx}{(tan^{-1}x)(1+x^{2})}+\dfrac{dy}{y}=0$$
let $$tan^{-1}x=t$$
$$\Rightarrow \dfrac{1}{1+x^{2}}dx=dt$$
$$\Rightarrow \int \dfrac{dt}{t}+\int \dfrac{dy}{y}=log\ c$$
$$\Rightarrow log\ yt=log\ c$$
$$\Rightarrow yt=c$$
$$\Rightarrow y\ tan^{-1}x=c$$

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Re-Attempt Weekly Quiz Competition

Differential Equations Test - 21

Result

Differential Equations Test - 21

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SHARING IS CARING

Question 1

$$y=(x+2)^{2}-13log\left | x+2 \right |+c$$

$$y=(x+2)^{2}-5log\left | x+2 \right |+c$$

$$y=\frac{x^{2}}{2}+2x+13log\left | x+2 \right |+c$$

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

Solution

Question 2

$$(y+1)(1+e^{-x})=ce^{y}$$

$$(e^{x}+1)y=c$$

$$(1+e^{x})(y+1)=c$$

$$(e^{x}+1)x=c$$

Solution

Question 3

$$x+y+\dfrac{1}{x}+\dfrac{1}{y}=c$$

$$2(x+y)-\left ( \dfrac{1}{x}+\dfrac{1}{y} \right )=c$$

$$2(x-y)+\left ( \dfrac{1}{x}-\dfrac{1}{y} \right )=c$$

$$(x-y)-\left ( \dfrac{1}{x}+\dfrac{1}{y} \right )=c$$

Solution

Question 4

$$y=log(e^{3x}+x^{2})+c$$

$$y=2log(e^{3x}+x^{2})+c$$

$$2e^{2x}=3(e^{3y}+y^{3})+c$$

$$3e^{2y}=2(e^{3x}+x^{3})+c$$

Solution

Question 5

$$log\left | y \right |=sin^{2}x$$

$$log\left | y \right |=sinx$$

$$log\left | y \right |=sin2x$$

$$log\left | y \right |=cos^{2}x$$

Solution

Question 6

$$e^{\sqrt{y-1}}=cx$$

$$\sqrt{y-x}=ex$$

$$\dfrac{1}{\sqrt{y-1}}=cx$$

$$x(\sqrt{y-x})=c$$

Solution

Question 7

$$xy=x+y+c$$

$$x-y\dfrac{x}{y}+c=0$$

$$xy-x+y=c$$

$$x+y\dfrac{x}{y}+c=0$$

Solution

Question 8

$$sin^{-1}\left ( \dfrac{y}{2} \right )+\dfrac{1}{x}=c$$

$$\dfrac{x^{3}}{3}+sin^{-1}\left ( \dfrac{y}{4} \right )=c$$

$$\sin^{-1}\left ( \dfrac{y}{2} \right )+2x=c$$

$$\sqrt{4-y^{2}}+2x=c$$

Solution

Question 9

$$\tan x \tan y =c$$

$$y \tan y =x \tan x + c$$

$$\tan x \cos y = \tan y \cos x +c$$

$$y \tan y - x \tan x + log \left( \dfrac{cosy}{cosx} \right )=c$$

Solution

Question 10

$$ytan^{-1}x=c$$

$$xtan^{-1}x=0$$

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

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

Solution

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