Question

given a function 'g' which has a derivative g'(x) for every real 'x' and which satisfy g'(0)=2 and g(x+y)=e^y*g(x) + e^x*g(y) foe all x & y. Find g(x) pl. help immediate medals awarded

Answer 1

maybe this would help f'(x)=lim h->0 [f(x+h)-f(x)]/h

Answer 2

\[g(x+y)=e^yg(x) + e^xg(y)\]
Differentiate
\[g'(x+y) \times (1+ \frac{dy}{dx})=e^y g'(x)+e^yg(x) \frac{dy}{dx}+e^x g'(y) \frac{dy}{dx}+e^x g(y) \}\]
\[\frac{dy}{dx}=0\]
\[g'(x+y)=e^y g'(x)+e^x g(y)\]
Replace y with x and x with 0
\[g'(x)=e^x g'(0)+e^0 g(x)\]
\\[\LARGE g(x)=g'(x)-2\]
Replace x with 0
\[g(0)=0\]
hmm..gotta think about rest,just tried :|

Answer 3

can there be a more simplified answer?curious :o

Answer 4

@rishabhjaiswal225499 what is the answer?

Answer 5

the answer 2x(e^(2x))

Answer 6

did u get what i did?

Answer 7

yup i got but couldt not do further

Answer 8

Differentiate again
\[g''(x)=g'(x)\]
Do you know any function whose differential is it itself?

Answer 9

It is \(ce^x\)

Answer 10

ok got it now

Answer 11

thanks