Question

If df(x)/dx=f(x) and f(1)=2 then f(3) is equal to

Answer 1

If df(x)/dx=f(x) and f(1)=2 then f(3) is equal to 2e²

Step-by-step explanation:

Given:

$\frac{df(x)}{dx}=f(x)$

And $f(1)=2$

To find out:

f(3)=?

Solution:

$\frac{df(x)}{dx}=f(x)$

$\implies \frac{df(x)}{f(x)}=dx$

$\implies \int \frac{df(x)}{f(x)}=\int dx$

$\implies \ln[f(x)]=x+c$       where c is an arbitrary constant

$\implies f(x)=e^{x+c}$

$\implies f(x)=e^x.e^c$

$\implies f(x)=Ce^x$

$f(1)=2$

$\implies 2=Ce^1$

$\implies C=\frac{2}{e}$

Now

$f(3)=Ce^3$

$\implies f(3)=\frac{2}{e}\times e^3$

$\implies f(3)=2e^2$

Hope this answer is helpful.

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