Differential Equations Test

Result

Differential Equations Test - 12

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

$$x^{\frac{b-c}{bc}} . x^{\frac{c-a}{ca}} . x^{\frac{a-b}{ac}}=$$

A
$$a^{a+b+c}$$

B
$$x^{abc}$$

C
1

D
0

Solution

$$x^{\cfrac{b-c}{bc}}$$$$.x^{\cfrac{c-a}{ac}}$$$$.x^{\cfrac{a-b}{ac}}$$
$$x^{\cfrac{b-c}{bc}+\cfrac{c-a}{ca}+\cfrac{a-b}{ac}}$$
$$x^0=1$$
Question 2
1 / -0

The number of arbitrary constants in the particular solution of a differential equation of third order are:

A
$$3$$

B
$$2$$

C
$$1$$

D
$$0$$

Solution

$$\textbf{Step 1: Find the number of arbitrary constants}$$
$$\text{In the particular solution of a differential equation of third order, there is no arbitrary}$$
$$\text{constant because, in the particular solution of any differential equation,}$$
$$\text{we remove all the arbitrary constants by substituting some particular values.}$$
$$\text{Arbitrary constants in the particular solution of a differential equation of third order is0}$$

$${\textbf{Hence, Option D is the answer}}$$
Question 3
1 / -0

Check whether the function is homogenous or not. If yes then find the degree of the function
$$g(x)=8x^4$$.

A
Homogenous with degree 4

B
Homogenous with degree 1

C
Not homogenous

D
None of these

Solution

Given function is $$g\left( x \right) =8{ x }^{ 4 }$$
To check the homogenity of the function we find $$g\left( kx \right) $$ which is,
$$g\left( kx \right) =8{ k }^{ 4 }{ x }^{ 4 }$$
$$\cfrac { g\left( kx \right) }{ g\left( x \right) } =\cfrac { 8{ k }^{ 4 }{ x }^{ 4 } }{ 8{ x }^{ 4 } } ={ k }^{ 4 }$$
$$\therefore$$ The function is homogenous with $$k>0$$ and is of degree $$4$$.
Question 4
1 / -0

What is the solution of the differential equation $$\dfrac {dx}{dy} + \dfrac {x}{y} - y^{2} = 0$$?
where $$c$$ is an arbitrary constant.

A
$$xy = x^{4} + c$$

B
$$xy = y^{4} + c$$

C
$$4xy = y^{4} + c$$

D
$$3xy = y^{3} + c$$

Solution

$$\dfrac { dx }{ dy } +\dfrac { x }{ y } -{ y }^{ 2 }=0\\ \Rightarrow \dfrac { dx }{ dy } =\dfrac { { y }^{ 3 }-x }{ y } \\ \Rightarrow ydx={ y }^{ 3 }dy-xdy\\ \Rightarrow xdy+ydx={ y }^{ 3 }dy\\ \Rightarrow d(xy)=d\left(\dfrac { { y }^{ 4 } }{ 4 }\right)$$
Integrating both the sides, we get
$${ y }^{ 4 }+c=4xy$$
Question 5
1 / -0

What is the solution of the differential equation $$\dfrac {ydx - xdy}{y^{2}} = 0$$?
where $$c$$ is an arbitrary constant.

A
$$xy = c$$

B
$$y = cx$$

C
$$x + y = c$$

D
$$x - y = c$$

Solution

$$\dfrac { ydx-xdy }{ { y }^{ 2 } } =0\\ \Rightarrow ydx=xdy$$
$$\Rightarrow \dfrac { dx }{x } =\dfrac { dy }{ y } $$
Integrating both the sides, we get
$$\displaystyle \int \dfrac { dx }{x } =\int \dfrac { dy }{ y } $$
$$\Rightarrow \ln(cx)=\ln(y)\\ \Rightarrow y=cx$$
where $$c$$ is an arbitrary constant.
Question 6
1 / -0

What is the solution of the differential equation $$\sin \left (\dfrac {dy}{dx}\right ) - a = 0$$?
where $$c$$ is an arbitrary constant.

A
$$y = x \sin^{-1} a + c$$

B
$$x = y \sin^{-1} a + c$$

C
$$y = x + x \sin^{-1} a + c$$

D
$$y = \sin^{-1} a + c$$

Solution

$$\sin\left( \dfrac { dy }{ dx } \right) =a$$
$$ \Rightarrow \dfrac { dy }{ dx } ={\sin }^{ -1 }a$$
$$\Rightarrow dy={\sin }^{ -1 }a\ dx$$
Integrating both the sides, we get
$$\displaystyle \int dy=\int {\sin }^{ -1 }a\ dx$$
$$\Rightarrow y=x\ {\sin }^{ -1 }a+c$$
where $$c$$ is a constant.
Question 7
1 / -0

Check whether the function is homogenous or not. If yes then find the degree of the function
$$g(x)=4-x^2$$.

A
Homogenous with degree 2

B
Not homogenous

C
Homogenous with degree 4

D
None of these

Solution

Given function is $$g\left( x \right) =4-{ x }^{ 2 }$$
To check the Homogenity of the function we find $$g\left( kx \right) $$
$$g\left( kx \right) =4-{ k }^{ 2 }{ x }^{ 2 }$$
$$\cfrac { g\left( kx \right) }{ g\left( x \right) } =\cfrac { 4-{ k }^{ 2 }{ x }^{ 2 } }{ 4-{ x }^{ 2 } } \neq { k }^{ n }$$
Where $$k>0$$and $$n$$ is an integer.
$$\therefore$$ The function is not homogenous.
Question 8
1 / -0

Which of the following is true regarding the function $$f(x, y)= x^4 \sin \dfrac{x}{y}$$ ?

A
Not homogenous

B
Homogenous with degree 2

C
Homogenous with degree 3

D
None of the above

Solution

given function is $$f\left( x,y \right) ={ x }^{ 4 }\sin { \cfrac { x }{ y } } $$
Now,$$f\left( kx,ky \right) ={ k }^{ 4 }{ x }^{ 4 }\sin { \cfrac { x }{ y } } $$ ,
$$\cfrac { f\left( kx,ky \right) }{ f\left( x,y \right) } ={ k }^{ 4 }$$
$$\therefore$$ the function is homogenous with Degree $$4$$.
Question 9
1 / -0

What is the solution of the differential equation $$\dfrac{dy}{dx} +\dfrac{y}{x} = 0 $$ ?
Where c is a constant.

A
$$xy = c$$

B
$$x = cy$$

C
$$y = cx$$

D
None of the above

Solution

Given expression is
$$\dfrac { dy }{ dx } +\dfrac { y }{ x } =0\\ \Rightarrow \dfrac { dy }{ dx } =-\dfrac { y }{ x } \\ \Rightarrow \dfrac { dy }{ y } +\dfrac { dx }{ x } =0$$
Integrating we get
$$\displaystyle \int { \dfrac { dy }{ y } } +\int { \frac { dx }{ x } } =0\\ \Rightarrow \ln { y } +\ln { x=c } \\ \Rightarrow xy=c$$
Question 10
1 / -0

Find the value of $$k$$ for the function: $$2x^2y+3xyz+z^k$$ to be homogenous.

A
$$6$$

B
$$3$$

C
$$2$$

D
None of these

Solution

Let the given function be $$f\left( x,y,z \right) =2{ x }^{ 2 }y+3xyz+{ z }^{ k }$$
Given that the function is homogenous,
$$\Longrightarrow \cfrac { f\left( \alpha x,\alpha y,\alpha z \right) }{ f\left( x,y,z \right) } =\cfrac { 2{ x }^{ 2 }y\alpha ^{ 3 }+3xyz{ \alpha }^{ 3 }+{ z }^{ k }{ \alpha }^{ k } }{ 2{ x }^{ 2 }y+3xyz+{ z }^{ k } } ={ \alpha }^{ n }$$
where $$\alpha $$ is the degree of he function and $$n$$ is a positive integer.
$$\therefore$$ by comparing the terms in the numerator we get $$k=3$$
so that we can take common in the numerator such that it cancels with the denominator.

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Differential Equations Test - 12

Result

Differential Equations Test - 12

Score

-

Rank

-

SHARING IS CARING

Question 1

$$a^{a+b+c}$$

$$x^{abc}$$

1

0

Solution

Question 2

$$3$$

$$2$$

$$1$$

$$0$$

Solution

Question 3

Homogenous with degree 4

Homogenous with degree 1

Not homogenous

None of these

Solution

Question 4

$$xy = x^{4} + c$$

$$xy = y^{4} + c$$

$$4xy = y^{4} + c$$

$$3xy = y^{3} + c$$

Solution

Question 5

$$xy = c$$

$$y = cx$$

$$x + y = c$$

$$x - y = c$$

Solution

Question 6

$$y = x \sin^{-1} a + c$$

$$x = y \sin^{-1} a + c$$

$$y = x + x \sin^{-1} a + c$$

$$y = \sin^{-1} a + c$$

Solution

Question 7

Homogenous with degree 2

Not homogenous

Homogenous with degree 4

None of these

Solution

Question 8

Not homogenous

Homogenous with degree 2

Homogenous with degree 3

None of the above

Solution

Question 9

$$xy = c$$

$$x = cy$$

$$y = cx$$

None of the above

Solution

Question 10

$$6$$

$$3$$

$$2$$

None of these

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-

Results Announcement on: coming Monday

Stay Tuned: We’ll update you on your result via email or SMS.