Maths-
General
Easy

Question

If f left parenthesis x right parenthesis space equals space f apostrophe left parenthesis x right parenthesis and f(1) = 2, then f(3) =

  1. straight e squared
  2. 2 straight e squared
  3. 3 straight e squared
  4. 2 straight e cubed

hintHint:

In the given question we have to find f(3) for this we will find the integrating factor and then find the solution.

The correct answer is: 2 straight e squared


    G i v e n comma space f left parenthesis x right parenthesis equals f apostrophe left parenthesis x right parenthesis comma space f left parenthesis 1 right parenthesis equals 2
l e t space f left parenthesis x right parenthesis equals y
rightwards double arrow f apostrophe left parenthesis x right parenthesis equals fraction numerator d y over denominator d x end fraction
rightwards double arrow y equals fraction numerator d y over denominator d x end fraction
rightwards double arrow fraction numerator d y over denominator d x end fraction minus y equals 0
T h i s space i s space a space l i n e a r space d i f f e r e n t i a l space e q u a t i o n comma fraction numerator d y over denominator d x end fraction plus P y equals Q
P equals negative 1 space Q equals 0
I F equals e to the power of integral P d x end exponent equals e to the power of integral negative 1 d x end exponent equals e to the power of negative x end exponent
S o l u t i o n space o f space d i f f e r e n t i a l space e q u a t i o n
y space I. F equals integral Q. space I F space d x plus C
y. e to the power of negative x end exponent equals integral 0. e to the power of negative x end exponent d x plus C
y. e to the power of negative x end exponent equals C
f left parenthesis x right parenthesis. e to the power of negative x end exponent equals C
g i v e n comma space f left parenthesis 1 right parenthesis equals 2
f left parenthesis 1 right parenthesis. e to the power of negative 1 end exponent equals 2
f left parenthesis 1 right parenthesis equals 2 over e rightwards double arrow C equals 2 over e
f left parenthesis 3 right parenthesis. e to the power of negative 3 end exponent equals 2 over e
f left parenthesis 3 right parenthesis equals 2 over e e cubed equals 2 e squared

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