Differential Equations Test

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Differential Equations Test - 45

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Weekly Quiz Competition

Question 1
1 / -0

if $$ y = y(x) $$ and $$ \dfrac{ 2 + sinx }{y + 1}(\dfrac{dy}{dx}) = -cosx, y(0) = 1, then y(\pi/2) $$ equals

A
1/3

B
2/3

C
-1/3

D
1

Solution

$$ \dfrac{1}{y + 1}dy = -\dfrac{cosx}{2 + sinx}dx $$
Integrating, we get
$$ log (y + 1) + log k + log(2+sinx) = 0 $$
$$ \therefore k(y + 1)(2 + sinx) = 1 $$ when $$ x=0, y=1 $$ where $$ k $$ is constant.
$$ \therefore 4k = 1 $$ or $$ k =1/4 $$
$$\therefore (y+ 1)(2 + sinx) = 4 $$
Now put $$ x = \pi/2 \therefore (y + 1)3 = 4 $$
$$ \therefore y = \dfrac{1}{3} $$
Question 2
1 / -0

Let $$f(x)$$ be a function such that $$f(0)=f'(0)=0, f''(x)=\sec^{4}x+4$$, then the function is

A
$$\log (\sin x)+\dfrac{1}{3}\tan^{3}x+xb$$

B
$$\dfrac{2}{3}\log (\sec^2 x)+\dfrac{1}{6}\tan^{2}x+2x^{2}$$

C
$$\log (\cos x)+\dfrac{1}{6}\cos^{2}x+\dfrac{x^{2}}{5}$$

D
$$None\ of\ these$$

Solution
Question 3
1 / -0

The solution of the equation $$ (x^{2}y + x^{2})dx + y^{2}(x-1)dy = 0 $$ is given by

A
$$ x^{2} + y^{2} + 2(x-y) + 2 ln \dfrac{(x-1)(y+1)}{c} = 0 $$

B
$$ x^{2} + y^{2} + 2(x-y) + ln \dfrac{(x-1)(y+1)}{c} = 0 $$

C
$$ x^{2} + y^{2} + 2(x-y) - 2 ln \dfrac{(x-1)(y+1)}{c} = 0 $$

D
None of these

Solution

$$ x^{2}(y+1)dx + y^{2}(x-1)dy = 0 $$
$$ \implies \dfrac{x^{2}dx}{x-1} = -\dfrac{y^{2}dy}{y+1} $$
$$ \implies \int [x+1+\dfrac{1}{x-1}]dx = -\int[y-1+\dfrac{1}{y+1}]dy $$
$$ \implies \dfrac{x^{2}}{2} + x + ln(x-1) = -[\dfrac{y^{2}}{2} - y + ln(y+1)] + lnc $$
$$ \implies \dfrac{x^{2} + y^{2}}{2} + (x-y) + ln(\dfrac{(x-1)(y+1)}{c} = 0 $$
Question 4
1 / -0

Solution of $$ 2y sin x \frac {dv}{dx}= 2 sin x cos x -y^2 cos x, $$ for $$x = \frac { \pi}{2} , y = 1 $$ is

A
$$ y^2 = sin x $$

B
$$ y = sin^2 x $$

C
$$ y^2 = 1 +cos x $$

D
None of these

Solution
Question 5
1 / -0

If $$ \phi(x) =\int \left\{ \phi (x) \right\}^{-2} $$dx and $$ \phi ( 1) =0 $$ then $$ \phi (x) = $$

A
$$ \left\{ 2(x -1) \right\}^{1/4} $$

B
$$ \left\{ 5(x -2) \right\}^{1/5} $$

C
$$ \left\{ 3(x -1) \right\}^{1/3} $$

D
None of these

Solution
Question 6
1 / -0

If $$ x\dfrac{dy}{dx} = y(\log y - \log x + 1) $$, then the solution of the equation is

A
$$ \log \dfrac{x}{y} = cy $$

B
$$ \log \dfrac{y}{x} = cy $$

C
$$ \log \dfrac{x}{y} = cx $$

D
None of these

Solution

$$ \dfrac{dy}{dx} = \dfrac{x}{y}[\log \dfrac{y}{x} +1] $$
Put $$ y = vx $$
$$ v + x\dfrac{dy}{dx} = v\log v = v $$
$$ \therefore \dfrac{dv}{v \log v} = \dfrac{dx}{x} $$
$$ \therefore \log (\log v) = \log x + \log c = \log cx $$
$$ \therefore log \dfrac{y}{x} = cx $$
Question 7
1 / -0

Solution of $$ \dfrac{dy}{dx} + 2xy = y $$ is

A
$$ y = ce^{x-x^{2}} $$

B
$$ y = ce^{x^{2}-x} $$

C
$$ y = ce^{x} $$

D
$$ y = ce^{-x^{2}} $$

Solution

$$ \dfrac{dy}{dx} + 2xy = y $$
$$ \implies \dfrac{dy}{dx} = y(1 - 2x) $$
$$ \implies \dfrac{dy}{y} = (1-2x)dx $$
$$ \implies log y = x - x^{2} + c_{1} $$
$$ \implies y = e^{x - x^{2}} e^{c_{1}} = ce^{x - x^{2}} $$ where $$ c = e^{c_{1}} $$
$$ \implies y = ce^{x - x^{2}} $$ is the required solution.
Question 8
1 / -0

The solution of differential equation $$ yy' = x\big(\dfrac{y^{2}}{x^{2}} + \dfrac{f (y^{2}/x^{2})}{f' (y^{2}/x^{2})}\big) $$ is

A
$$ f(y^{2}/x^{2}) = cx^{2} $$

B
$$ x^{2}f(y^{2}/x^{2}) = c^{2}y^{2} $$

C
$$ x^{2}f(y^{2}/x^{2}) = c $$

D
$$ f(y^{2}/x^{2}) = cy/x $$

Solution

The given equation can be written as
$$ \dfrac{dy}{dx} = \big[\dfrac{y^{2}}{x^{2}} + \dfrac{f(y^{2}/x^{2})}{f'(y^{2}/x^{2})}\big] $$
Above equation is a homogeneous equation.
Putting $$ y = vx $$, we get
$$ v [v + x\dfrac{dv}{dx}] = v^{2} + \dfrac{f(v^{2})}{f'(v^{2})} $$
$$ \implies vx\dfrac{dv}{dx} = \dfrac{f(v^{2})}{f'(v^{2})} $$ variable separable
$$ \implies \dfrac{2vf'(v^{2})}{f(v^{2})}dv = 2\dfrac{dv}{x} $$
Now integrating both side, we get
$$ \log f(v^{2}) = \log x^{2} + \log c $$ [$$ \log c = $$ constant]
or $$ f(v^{2}) = \log cx^{2} $$
or $$ f(y^{2}/x^{2}) = cx^{2} $$
Question 9
1 / -0

if integrating factor of $$ x(1-x^{2})dy + (2x^{2}y - y -ax^{3})dx=0 $$ is $$ e^{\int pdx} $$, then P is equal to

A
$$ \dfrac{2x^{2} - ax^{3}}{x(1-x^{2})} $$

B
$$ 2x^{3} - 1 $$

C
$$ \dfrac{2x^{2} - a}{ax^{3}} $$

D
$$ \dfrac{2x^{2} - 1}{x(1-x^{2})} $$

Solution

$$ x(1-x^{2})dy + (2x^{2}y-y-ax^{3})dx = 0 $$
$$ \implies x(1-x^{2}) \dfrac{dy}{dx} + 2x^{2}y-y-ax^{3} = 0 $$
$$ \implies x(1-x^{2}) \dfrac{dy}{dx} + y(2x^{2}-1) = ax^{3}) $$
$$ \implies \dfrac{dy}{dx} + \dfrac{2x^{2}-1}{x(1-x^{2})}y= \dfrac{ax^{3}}{x(1-x^{2})} $$
which is of the form $$ \dfrac{dy}{dx} +Py = Q $$
Its integrating factor is $$ e^{\int Pdx} $$
Here $$ P=\dfrac{2x^{2}}{x(1-x^{2})} $$
Question 10
1 / -0

The solution of differentiation equation $$(2y+xy^{3})dx+(x+x^{2}y^{2})dy=o $$ is

A
$$ x^{2}y+\dfrac{x^{3}y^{3}}{3}= c $$

B
$$ xy^{2}+\dfrac{x^{3}y^{3}}{3}= c $$

C
$$ x^{2}y+\dfrac{x^{4}y^{4}}{4}= c $$

D
None of these

Solution

$$ (2y+xy^{3})dx+(x+x^{2}y^{2})dy = 0 $$
$$ \implies (2ydx + xdy) + (xy^{3}dx+x^{2}y^{2}dy) + 0 $$
Multiplying by $$x$$, we get
$$ (2xydx + x^{2}dy) + (x^{2}y^{3}dx+x^{3}y^{2}dy) = 0 $$
$$ \implies d(x^{2}y)+\dfrac{1}{3}d(x^{3}y^{3}) = 0 $$
Integrating, we get $$ x^{2}y + \dfrac{x^{3}y^{3}}{3} = c $$

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Re-Attempt Weekly Quiz Competition

Differential Equations Test - 45

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Differential Equations Test - 45

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SHARING IS CARING

Question 1

1/3

2/3

-1/3

1

Solution

Question 2

$$\log (\sin x)+\dfrac{1}{3}\tan^{3}x+xb$$

$$\dfrac{2}{3}\log (\sec^2 x)+\dfrac{1}{6}\tan^{2}x+2x^{2}$$

$$\log (\cos x)+\dfrac{1}{6}\cos^{2}x+\dfrac{x^{2}}{5}$$

$$None\ of\ these$$

Solution

Question 3

$$ x^{2} + y^{2} + 2(x-y) + 2 ln \dfrac{(x-1)(y+1)}{c} = 0 $$

$$ x^{2} + y^{2} + 2(x-y) + ln \dfrac{(x-1)(y+1)}{c} = 0 $$

$$ x^{2} + y^{2} + 2(x-y) - 2 ln \dfrac{(x-1)(y+1)}{c} = 0 $$

None of these

Solution

Question 4

$$ y^2 = sin x $$

$$ y = sin^2 x $$

$$ y^2 = 1 +cos x $$

None of these

Solution

Question 5

$$ \left\{ 2(x -1) \right\}^{1/4} $$

$$ \left\{ 5(x -2) \right\}^{1/5} $$

$$ \left\{ 3(x -1) \right\}^{1/3} $$

None of these

Solution

Question 6

$$ \log \dfrac{x}{y} = cy $$

$$ \log \dfrac{y}{x} = cy $$

$$ \log \dfrac{x}{y} = cx $$

None of these

Solution

Question 7

$$ y = ce^{x-x^{2}} $$

$$ y = ce^{x^{2}-x} $$

$$ y = ce^{x} $$

$$ y = ce^{-x^{2}} $$

Solution

Question 8

$$ f(y^{2}/x^{2}) = cx^{2} $$

$$ x^{2}f(y^{2}/x^{2}) = c^{2}y^{2} $$

$$ x^{2}f(y^{2}/x^{2}) = c $$

$$ f(y^{2}/x^{2}) = cy/x $$

Solution

Question 9

$$ \dfrac{2x^{2} - ax^{3}}{x(1-x^{2})} $$

$$ 2x^{3} - 1 $$

$$ \dfrac{2x^{2} - a}{ax^{3}} $$

$$ \dfrac{2x^{2} - 1}{x(1-x^{2})} $$

Solution

Question 10

$$ x^{2}y+\dfrac{x^{3}y^{3}}{3}= c $$

$$ xy^{2}+\dfrac{x^{3}y^{3}}{3}= c $$

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

None of these

Solution

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