COMEDK 2024 Shift 1 Paper

The given differential equation f(x)=f′(x) implies that the derivative of the function is equal to the function itself. This is a characteristic property of exponential functions. Let's solve the differential equation to find the general solution:
We can rewrite the equation as ‌

df

dx

=f(x)
Separating the variables, we get:

‌

df

f(x)

=dx
Integrating both sides:
∫‌

df

f(x)

=∫dx
This gives us:
ln|f(x)|=x+C

where C is the constant of integration.
Solving for f(x) :
f(x)=ex+C=eC⋅ex
Since eC is just another constant, we can write the general solution as:

f(x)=Aex
where A is an arbitrary constant.
Now, we use the initial condition f(1)=2 to find the value of A :
‌2=Ae1
‌A=2∕e

Therefore, the specific solution to the differential equation is:
f(x)=‌

2

e

ex=2ex−1
Finally, we can find f(3) by substituting x=3 :
f(3)=2e3−1=2e2
Therefore, the answer is Option C: 2e2