Differential Equations Test

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Differential Equations Test - 13

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Weekly Quiz Competition

Question 1
1 / -0

If $$\dfrac{dy}{dx}=x^{-3}$$ then $$y$$

A
$$-\dfrac{1}{2x^2}+c$$

B
$$\dfrac{1}{2x^2}+c$$

C
$$-\dfrac{1}{3x^3}+c$$

D
$$-\dfrac{1}{4x^4}+c$$

Solution

$$\displaystyle{\frac { dy }{ dx } ={ x }^{ -3 }\\ dy={ x }^{ -3 }dx\\ \int { dy } =\int { { x }^{ -3 }dx } \\ y=\frac { { x }^{ -2 } }{ -2 } +c\\ y=-\frac { 1 }{ 2{ x }^{ 2 } } +c}$$
Question 2
1 / -0

The number of arbitrary constants in the particular solution of the differential equation of order $$3$$ is ______.

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

The number of arbitary constants in the particular solution of the differential equation of order $$3$$ is $$0$$
Question 3
1 / -0

The number of arbitrary constant in the general solution of differential equation of order $$3$$ is _________.

A
$$0$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution

As per the note given the number of arbitrary constants in third order differential equation are $$3$$(Three).
Question 4
1 / -0

What is the general solution of the differential equation $$x\, dy - y\, dx \,y^2$$ ?

A
$$x = cy $$

B
$$y^2=cx$$

C
$$x + xy - cy = 0$$

D
None of the above

Solution

$$xdy-{ y }^{ 3 }dx=0\\ rearranging\quad \\ \cfrac { dx }{ x } =\cfrac { dy }{ { y }^{ 3 } } \\ integrating\quad both\quad sides\\ \ln { x } =\frac{{ y }^{ -2 }}{-2}+c\\ $$
Question 5
1 / -0

The solution of the differential equation $$x^4\dfrac{dy}{dx}+x^3y+cosec(xy)=0$$ is equal to

A
$$2 \cos (xy)+x^{-2}=C$$

B
$$2\cos(xy)+y^{-2}=C$$

C
$$2\sin(xy)+x^{-2}=C$$

D
$$2\sin (xy)+y^{-2}+C$$

Solution

Given,
$$x^4\dfrac{dy}{dx}+x^3y+cosec(xy)=0$$

Put $$xy=t$$
$$y+x\dfrac{dy}{dx}=\dfrac{dt}{dx}$$

Substituting it we get;
$$x^3\left(\dfrac{dt}{dx}-\dfrac{t}{x}\right)+x^2t+cosec(t)=0$$

$$\Rightarrow x^3\dfrac{dt}{dx}+cosec(t)=0$$

$$\Rightarrow \sin(t)dt+x^{-3}dx=0$$

After Integrating we get;

$$2\cos(xy)+x^{-2}=C$$
Question 6
1 / -0

The differential equation $$y\cfrac { dy }{ dx } +x=a$$ where '$$a$$' is any constant represents:

A
A set of straight lines

B
A set of ellipses

C
A set of circles

D
None of the above

Solution

$$y\frac { dy }{ dx } +x=a$$
$$\Rightarrow ydy+xdx=adx$$
Integrating both sides,
$$\Rightarrow \dfrac { { y }^{ 2 } }{ 2 } +\dfrac { { x }^{ 2 } }{ 2 } =ax+c\\ \Rightarrow { x }^{ 2 }+{ y }^{ 2 }-2ax-2c=0\\ \Rightarrow { (x-a) }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }+2c$$
Clearly, this equation represents circle with centre $$(a,0)$$ and radius $$\sqrt { { a }^{ 2 }+2c }. $$
Option C is correct.
Question 7
1 / -0

The differential equation of $$y=c{ x }^{ 3 },c$$ is a arbitrary constant is _______

A
$$x.\cfrac { dy }{ dx } =3y$$

B
$${ x }^{ 3 }.\cfrac { dy }{ dx } =3y$$

C
$$3x\cfrac { dy }{ dx } =y$$

D
$$x.\cfrac { dy }{ dx } =3{ y }^{ 2 }$$

Solution

$$y=cx^{3}$$
Differentiating both sides
$$\dfrac{dy}{dx}=\dfrac{d}{dx}(cx^{3})$$
$$\dfrac{dy}{dx}=c\dfrac{d}{dx}(x^{3})$$
$$\dfrac{dy}{dx}=3cx^{2}$$
Multiplying x on both sides
$$x\dfrac{dy}{dx}=x\times 3cx^{2}$$
$$x\dfrac{dy}{dx}=3cx^{3}$$
Substituting $$y=cx^{3}$$
$$x\dfrac{dy}{dx}=3y$$
Question 8
1 / -0

The solution of $$\dfrac{ydx-xdy}{y^{2}}=0$$ represents a family of

A
Straight lines passing through the origin

B
Circles

C
Parabola

D
Hyperbola

Solution

$$\dfrac{{ydx - xdy}}{{{y^2}}} = 0$$
$$ \Rightarrow d\left( {\dfrac{x}{y}} \right) = 0$$
integrating both sides
$$ \Rightarrow \dfrac{x}{y} = c$$
$$\Rightarrow x = cy$$
This is a straight passing through origin
Question 9
1 / -0

The $$D.E$$ $$y\dfrac { dy }{ dx } +x=a$$ represents

A
a circle whose centra is on $$X-$$axis

B
a circle whose centre is on $$Y-$$axis

C
a circle whose centre is origin

D
a parabola

Solution

$$y\frac{{dy}}{{dx}} + x = a$$
$$\Rightarrow ydy + xdx = adx$$
Integrating both sides
$$\Rightarrow \dfrac{{{y^2}}}{2} + \dfrac{{{x^2}}}{2} = ax + c$$
$$\Rightarrow {x^2} + {y^2} = 2ax + c$$
$$ \Rightarrow {x^2} + {y^2} - 2ax + c = 0$$
on comparing it with
$${x^2} + {y^2} + 2gx + + 2fy + c = 0$$
here , $$g = - a,f = 0$$
centre =$$\left( { - g, - f} \right) = \left( {a,0} \right)$$
therefore , it represent a circle with centre at X- axis
Question 10
1 / -0

The solution of $$\left( { x }^{ 2 }{ y }^{ 3 }+{ x }^{ 2 } \right) dx+\left( { y }^{ 2 }{ x }^{ 3 }+{ y }^{ 2 } \right) dy=0$$ is

A
$$\left( { x }^{ 3 }+1 \right) \left( { y }^{ 3 }+1 \right) =c$$

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

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

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

Solution

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

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Re-Attempt Weekly Quiz Competition

Differential Equations Test - 13

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Differential Equations Test - 13

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SHARING IS CARING

Question 1

$$-\dfrac{1}{2x^2}+c$$

$$\dfrac{1}{2x^2}+c$$

$$-\dfrac{1}{3x^3}+c$$

$$-\dfrac{1}{4x^4}+c$$

Solution

Question 2

$$0$$

$$1$$

$$2$$

$$3$$

Solution

Question 3

$$0$$

$$2$$

$$3$$

$$4$$

Solution

Question 4

$$x = cy $$

$$y^2=cx$$

$$x + xy - cy = 0$$

None of the above

Solution

Question 5

$$2 \cos (xy)+x^{-2}=C$$

$$2\cos(xy)+y^{-2}=C$$

$$2\sin(xy)+x^{-2}=C$$

$$2\sin (xy)+y^{-2}+C$$

Solution

Question 6

A set of straight lines

A set of ellipses

A set of circles

None of the above

Solution

Question 7

$$x.\cfrac { dy }{ dx } =3y$$

$${ x }^{ 3 }.\cfrac { dy }{ dx } =3y$$

$$3x\cfrac { dy }{ dx } =y$$

$$x.\cfrac { dy }{ dx } =3{ y }^{ 2 }$$

Solution

Question 8

Straight lines passing through the origin

Circles

Parabola

Hyperbola

Solution

Question 9

a circle whose centra is on $$X-$$axis

a circle whose centre is on $$Y-$$axis

a circle whose centre is origin

a parabola

Solution

Question 10

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

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

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

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

Solution

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