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The general solution of the differential equation

$(y^{2}-x^{3})dx-xydy=0\: \: (x\neq 0)$ is :

( where c is a constant of integration)

• Option 1)

$y^{2}-2x^{2}+cx^{3}=0$

• Option 2)

$y^{2}+2x^{3}+cx^{2}=0$

• Option 3)

$y^{2}+2x^{2}+cx^{3}=0$

• Option 4)

$y^{2}-2x^{3}+cx^{2}=0$

general solution of the differential equation  Option 1) Option 2) Option 3) Option 4)

Consider the differential equation, $y^{2}dx+\left ( x-\frac{1}{y} \right )dy=0.$ If value of y is 1 when $x=1,$ then the value of x for which $y=2,$ is :

• Option 1)

$\frac{1}{2}+\frac{1}{\sqrt{e}}$

• Option 2)

$\frac{5}{2}+\frac{1}{\sqrt{e}}$

• Option 3)

$\frac{3}{2}-\frac{1}{\sqrt{e}}$

• Option 4)

$\frac{3}{2}-\sqrt{e}$

IF=                when  Option 1) Option 2) Option 3)   Option 4)

If $e^{y}+xy=e,$ the ordered pair $\left ( \frac{\mathrm{d} y}{\mathrm{d} x},\frac{\mathrm{d}^{2}y }{\mathrm{d} x^{2}} \right )$ at $x=0$ is equal to :

• Option 1)

$\left ( -\frac{1}{e},-\frac{1}{e^{2}} \right )$

• Option 2)

$\left ( \frac{1}{e},-\frac{1}{e^{2}} \right )$

• Option 3)

$\left (- \frac{1}{e},\frac{1}{e^{2}} \right )$

• Option 4)

$\left ( \frac{1}{e},\frac{1}{e^{2}} \right )$

Again differentiate w.r.t.   x Option 1) Option 2)   Option 3)      Option 4)

Let $y=y(x)$ be the solution of the fifferential equation,

$\frac{\mathrm{d} y}{\mathrm{d} x}+ytanx=2x+x^{2}tanx,$ $x\epsilon (\frac{-\pi}{2},\frac{\pi}{2}),$

such that $y(0)=1$. Then :

• Option 1)

$y(\frac{\pi}{4})+y(\frac{-\pi}{4})=\frac{\pi^{2}}{2}+2$

• Option 2)

$y'(\frac{\pi}{4})+y'(\frac{-\pi}{4})=-\sqrt2$

• Option 3)

$y(\frac{\pi}{4})-y(\frac{-\pi}{4})=\sqrt2$

• Option 4)

$y'(\frac{\pi}{4})-y'(\frac{-\pi}{4})=\pi-\sqrt2$

C

If y = y(x) is the solution of the differential equation

$\frac{dy}{dx}=(tanx-y)sec^{2}x$

$x\epsilon (\frac{-\pi}{2},\frac{\pi}{2})$, such that y(0) = 0 , then

$y(\frac{-\pi}{4})$ is equal to :

• Option 1)

$e-2$

• Option 2)

$\frac{1}{2}-e$

• Option 3)

$2+\frac{1}{e}$

• Option 4)

$\frac{1}{e}-2$

Given differential eqn I.F. = =  Solution of given differential eqn                                  Given that  =>  put  So, correct option is (1) Option 1) Option 2) Option 3) Option 4)

If   $\cos x \frac{dy}{dx} - y \sin x = 6x, \left ( 0,

and  $y \left ( \frac{\pi}{3} \right )=0$ , then  $y \left ( \frac{\pi}{6} \right )$  is equal to :

• Option 1)

$\frac{\pi^{2}}{2\sqrt{3}}$

• Option 2)

$-\frac{\pi^{2}}{2}$

• Option 3)

$-\frac{\pi^{2}}{2\sqrt{3}}$

• Option 4)

$-\frac{\pi^{2}}{4\sqrt{3}}$

Option 1) Option 2) Option 3)   Option 4)

The solution of the differential equation $x\frac{dy}{dx}+2y=x^{2}\left ( x\neq 0 \right )$  with $y\left ( 1 \right )=1,$ is :

• Option 1)

$y=\frac{4}{5}x^{3}+\frac{1}{5x^{2}}$

• Option 2)

$y=\frac{x^{3}}{5}+\frac{1}{5x^{2}}$

• Option 3)

$y=\frac{x^{2}}{4}+\frac{3}{4x^{2}}$

• Option 4)

$y=\frac{3}{4}x^{2}+\frac{1}{4x^{2}}$

This D.E in the form of Linear D.E. Solution, given       Option 1)          Option 2) Option 3)        Option 4)

Given that the slope of the tangent to a curve $y=y(x)$ at any point $\left ( x,y \right )$ is $\frac{2y}{x^{2}}$. If the curve passes through the centre of the circle $x^{2}+y^{2}-2x-2y=0$, then its equation is :

• Option 1)

$x\log_{e}\left | y \right |=2(x-1)$

• Option 2)

$x\log_{e}\left | y \right |=-2(x-1)$

• Option 3)

$x^{2}\log_{e}\left | y \right |=-2(x-1)$

• Option 4)

$x\log_{e}\left | y \right |=x-1$

given slope of tangent  Integrate but side. It passes through centre of circle    Eq. of curve is  Option 1) Option 2) Option 3)   Option 4)

Let $y=y(x)$ be the solution of the differential equation, $(x^{2}+1)^{2}\frac{dy}{dx}+2x(x^{2}+1)y=1$ such that $y(0)=0.$ If $\sqrt{a}\; y\; (1)=\frac{\pi}{32}$, then the value of 'a' is :

• Option 1)

$\frac{1}{16}$

• Option 2)

$\frac{1}{4}$

• Option 3)

$1$

• Option 4)

$\frac{1}{2}$

this is a linear differential equation with We have   Option 1) Option 2) Option 3)   Option 4)

The solution of the differential equation,

$\frac{\mathrm{dy} }{\mathrm{d} x}= (x-y)^{2} , \: when \: y(1)=1,\: is:$

• Option 1)

$-\log_{e}\left | \frac{1+x-y}{1-x+y} \right | = x+y-2$

• Option 2)

$-\log_{e}\left | \frac{1-x+y}{1+x-y} \right | = 2(x-1)$

• Option 3)

$\log_{e}\left | \frac{2-y}{2-x} \right | = 2(y-1)$

• Option 4)

$\log_{e}\left | \frac{2-x}{2-y} \right | = x-y$

Solution of Differential Equation - put       - wherein Equation with convert to   let  y(1) = 1 or      Option 1)Option 2)Option 3)Option 4)

If    $y(x)$  is the solution of the differential equation $\frac{dy}{dx}+(\frac{2x+1}{x}) y =e^{-2x} , x> 0,$ where $y(1)=\frac{1}{2}e^{-2}$, then :

• Option 1)

$y(\log_{e}2)=\log_{e}4$

• Option 2)

$y(x)$  is decreasing in $(\frac{1}{2},1)$

• Option 3)

$y(\log_{e}2)=\frac{\log_{e}2}{4}$

• Option 4)

$y(x)$ is decreasing in (0,1)

Linear Differential Equation - - wherein P, Q are functions of x alone.     Linear Differential Equation - Multiply by   which is the Integrating factor - wherein P is the function of x alone   Solution of a differential equation, Hence,   if   is decreasing in       Option 1)  Option 2)    is decreasing in Option 3)  Option 4)  is decreasing in (0,1)

Let f:$R\rightarrow R$ be a function such that $f(x)= x^3+x^2{f}'(1)+x{f}''(2)+{f}'''(3)$,$x\epsilon R$. Then f(2) equals :

• Option 1)

-4

• Option 2)

30

• Option 3)

-2

• Option 4)

8

Differential Equations - An equation involving independent variable (x), dependent variable (y) and derivative of dependent variable with respect to independent variable  - wherein eg:        Order of a Differential Equation - The order of a differential equation is order of highest order occuring in differential equation - wherein order of    is...

If   $\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{3}{\cos ^2x }y=\frac{1}{\cos ^2x },x\epsilon \left ( \frac{-\pi }{3},\frac{\pi }{3} \right ),$ ,and $y\left ( \frac{\pi }{4} \right )= \frac{4}{3},$

then $y(-\pi/4)$ equals

• Option 1)

$1/3+e^6$

• Option 2)

1/3

• Option 3)

-4/3

• Option 4)

$1/3+e^3$

Linear Differential Equation - - wherein P, Q are functions of x alone.     Linear Differential Equation - Multiply by   which is the Integrating factor - wherein P is the function of x alone From the concept I.F. = or Now put  Option 1)  Option 2)  1/3Option 3)  -4/3Option 4)

The curve amongst the family of curves responded by the differential equation, $(x^2 - y^2)dx +2xy\: \: \: dy = 0$ which passes through (1,1), is:

• Option 1)

A hyperbola with the transverse axis along the x-axis

• Option 2)

A circle with centre on the x-axis

• Option 3)

an ellipse with the major axis along the y-axis.

• Option 4)

A circle with centre on the y-axis

Homogeneous Differential Equation - A function   is called homogeneous function of  degree n, if - wherein eg:     Homogeneous Differential Equation - Put - The D.E. can be written as  from the concept  put  On Integrating Passes through (1,1)        Option 1)A hyperbola with the transverse axis along the x-axisOption 2)A circle with centre on the x-axisOption 3)an ellipse with...

For $x>1,$ if $\left ( 2x \right )^{2y}=4e^{2x-2y}$,  then $\left ( 1+\log_{e}2x \right )^{2}\frac{\mathrm{d} y}{\mathrm{d} x}$  is equal to :

• Option 1)

$\frac{x\log_{e}2x+\log_{e}2}{x}$

• Option 2)

$x\log_{e}2x$

• Option 3)

$\log_{e}2x$

• Option 4)

$\frac{x\log_{e}2x-\log_{e}2}{x}$

Differential Equations - An equation involving independent variable (x), dependent variable (y) and derivative of dependent variable with respect to independent variable  - wherein eg:      Using   both sides     Option 1)       Option 2)Option 3)Option 4)

Let $y=y(x)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x\log_{e}x,\: \: \left ( x> 1 \right ).$ If $2y(2)=\log_{e}4-1,$ then $y(e)$ is equal to :

• Option 1)

$\frac{e^{2}}{4}$

• Option 2)

$-\frac{e}{2}$

• Option 3)

$-\frac{e^{2}}{2}$

• Option 4)

$\frac{e}{4}$

Linear Differential Equation - - wherein P, Q are functions of x alone.     Linear Differential Equation - Multiply by   which is the Integrating factor - wherein P is the function of x alone   Putting x = 2 we get C = 0 Option 1)      Option 2)Option 3)Option 4)

If $x= 3tant$ and y = 3sect, then the value of $\frac{d^2 y}{d x^2}$ at $t=\pi /4$,is :

• Option 1)

$3/2\sqrt{2}$

• Option 2)

$1/6\sqrt2$

• Option 3)

$1/3\sqrt{2}$

• Option 4)

1/6

Second Order Differential Equation - - wherein  Option 1)Option 2)Option 3)  Option 4)  1/6

Let $f:[0,1]\rightarrow R$ be such that $f(xy)=f(x)f(y)$, for all $x,y\epsilon [0,1]$,and $f(0)\neq 0$. If y = y(x) satisfies the differential equation, $\frac{\mathrm{d} y}{\mathrm{d} x}=f(x)$ with y(0) = 1, then $y(1/4)+y(3/4)$is equal to :

• Option 1)

3

• Option 2)

2

• Option 3)

5

• Option 4)

4

Differential Equations - An equation involving independent variable (x), dependent variable (y) and derivative of dependent variable with respect to independent variable  - wherein eg:      Given that At  Option 1)  3Option 2)  2Option 3)  5Option 4)  4

If $y = y(x)$ is the solution of the differential equation, $x\frac{dy}{dx} +2y = x^2$ satisfying $y (1) =1$ then $y\left(\frac{1}{2} \right )$ is equal to

• Option 1)

$\frac{7}{64}$

• Option 2)

$\frac{1}{4}$

• Option 3)

$\frac{49}{16}$

• Option 4)

$\frac{13}{16}$

Linear Differential Equation - - wherein P, Q are functions of x alone.     Linear Differential Equation - Multiply by   which is the Integrating factor - wherein P is the function of x alone Differential Equation can be written as. from the concept we have learnt  If =  Solution of this d.e is  So,  given  We need to find  Put   Option 1)Option 2)Option 3)Option 4)
Engineering
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At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers is given by . If the firm employs 25 more workers,then the new level of production of items is :

• Option 1)

4500

• Option 2)

2500

• Option 3)

3000

• Option 4)

3500

As we have learned Variable Separation Method - integrating, we get -   and    We have              Option 1) 4500 Option 2) 2500 Option 3) 3000 Option 4) 3500
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