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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,<x<\frac{\pi}{2} \right )

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 x is given by \frac{dp}{dx}=100-12\sqrt{x}. 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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