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  • I have a doubt, kindly clarify. - Let be such that, for all,and. If y = y(x) satisfies the differential equation, with y(0) = 1 - Differential equations - JEE Main

 

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

Answers (1)

best_answer

 

Differential Equations -

An equation involving independent variable (x), dependent variable (y) and derivative of dependent variable with respect to independent variable 
\left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )

- wherein

eg:

  \frac{d^{2}y}{dx^{2}}- 3\frac{dy}{dx}+5x=0

 

Given that

\\ f(xy) = f(x)\cdot f(y) \\ f(0) = 1 \;\textup{as}\; f(0 \neq 0 \Rightarrow f(x) = 1 \\ \frac{dy}{dx} = f(x) = 1\Rightarrow y = x +c

At x = 0, y = c \;\; i.e.\;\; y = 1\;so\;c =1

y = x + 1

y\left(\frac{1}{4} \right ) + y\left(\frac{3}{4} \right ) = \frac{1}{4} + 1 +\frac{3}{4} + 1 = 3


Option 1)

 

3

Option 2)

 

2

Option 3)

 

5

Option 4)

 

4

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