Differential Equations Test

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Differential Equations Test - 59

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Weekly Quiz Competition

Question 1
1 / -0

The solution of the equation $$\dfrac{dy}{dx}=\dfrac{(1+x)y}{(y-1)x}$$ is:

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

B
$$\log (\dfrac{x}{y})+x-y=c$$

C
$$y-x-\log xy=c$$

D
$$None\ of\ these$$

Solution
Question 2
1 / -0

Solution of $$\left(\dfrac {dy}{dx}\right)^{2}+x\dfrac {dy}{dx}-y=0$$ is

A
$$y=3x^{2}+9$$

B
$$y=3x+9$$

C
$$y=\dfrac {4}{3}x^{2}$$

D
$$y=9x+3$$

Solution
Question 3
1 / -0

Solution of $$\dfrac { d y } { d x } + 2 x y = y$$ is

A
$$y = c e ^ { x - x ^ { 2 } }$$

B
$$y = c e ^ { x ^ { 2 } } - x$$

C
$$y = c e ^ { x }$$

D
$$y = c e ^ { - x ^ { 2 } }$$

Solution
Question 4
1 / -0

The solution of $$x^2dy-y^2dx+xy^2(x-y)dy=0$$, is?

A
$$log\left|\dfrac{x-y}{xy}\right|=\dfrac{y^2}{2}+C$$

B
$$log\left|\dfrac{xy}{x-y}\right|=\dfrac{x^2}{2}+C$$

C
$$\log \left|\dfrac{x-y}{xy}\right|=\dfrac{x^2}{2}+C$$

D
$$log\left|\dfrac{x-y}{xy}\right|=x+C$$

Solution
Question 5
1 / -0

Let the population of rabbits surviving at a time t be governed by the differential equation $$\dfrac{dp(t)}{dt}=\dfrac{1}{2}p(t)-200.$$ If $$p(0)=100$$ , then p(t) equals:

A
$$600-500e^{t/2}$$

B
$$400-300e^{-t/2}$$

C
$$400-300e^{t/2}$$

D
$$300-200e^{-t/2}$$

Solution
Question 6
1 / -0

Solution of equation $$ \frac { d y } { d x } = \frac { y ^ { \frac { d } { d } ( \phi ( x ) ) } - y ^ { 2 } } { \phi ( x ) } $$ is

A
$$

y = \frac { \phi ( x ) + c } { x }

$$

B
$$

y = \frac { \phi ( x ) } { x } + c

$$

C
$$

y = \frac { \phi ( x ) } { x + c }

$$

D
$$

y = \phi ( x ) + x + c

$$

Solution
Question 7
1 / -0

An equation containing an independent variable, dependent variable, and differential coefficients of the dependent variable with respect to the independent variable is called a/an _______

A
Differential equation

B
Continuous function

C
Discontinuous function

D
Integral equation

Solution
Question 8
1 / -0

The integrating factor of the equation $$siny\dfrac{dy}{dx}=cosy\left(1-xcosy\right)$$ is

A
$$e^{-x}$$

B
$$e^{x}$$

C
$$e^{secy}$$

D
$$None\ of\ these$$

Solution
Question 9
1 / -0

Solution of differential equation $$\cfrac{dy}{dx}+x\sin^2{y}=\sin{y}\cos{y}$$ is-

A
$$\tan{y}=(x-1)+Ce-x$$

B
$$\cot{y}=(x-1)+Ce-x$$

C
$$\tan{y}=(x-1)ex+c$$

D
$$\cot{y}=(x-1)ex+c$$
Question 10
1 / -0

Consider the differential equation $$y^{ 2 }dx + \left(x - \dfrac { 1 }{ y }\right) dy = 0$$
If $$y \left( 1 \right) = 1$$, then $$x$$ is given by:

A
$$4 - \dfrac { 2 }{ y } - \dfrac {e^{\dfrac { 1 }{ y }}}e$$

B
$$3 - \dfrac { 1 }{ y } + \dfrac {e^{\dfrac { 1 }{ y }}}e$$

C
$$1 + \dfrac { 1 }{ y } - \dfrac {e^{\dfrac { 1 }{ y }}}e$$

D
$$1 - \dfrac { 1 }{ y } + \dfrac {e^{\dfrac { 1 }{ y }}}e$$

Solution