Help on :
if y = f(x) a function, satisfies the differential equation dy/dx = -4y and f(0) = 6, then f(x) =

Am I suppose to solve this using integration? If so, how am I suppose to do it??

Question

Help on :
if y = f(x) a function, satisfies the differential equation dy/dx = -4y and f(0) = 6, then f(x) = 

Am I suppose to solve this using integration? If so, how am I suppose to do it??

anonymous · Answer

I know that f(0) means that the C would be 6, but how do I solve the antiderivative of that thing?

anonymous · Answer

$$\frac{ dy }{ y }=-4dx$$
integrating 
$$\int\limits \frac{ 1 }{y }dy=-4 \int\limits dx+c$$
$$\ln y=-4x+c$$
when x=0,y=6
$$\ln 6=0+c,c=\ln 6$$
substitute the value of c and finally y=f(x)

anonymous · Answer

ah... so this is just like implicit differentiation, only the reverse?

anonymous · Answer

what I do to the one side I must do for the other to be equal, is that how we can derive the equation?? thanks

anonymous · Answer

but I think I am suppose to solve for y. Since y is in ln y, how do I find y again?

anonymous · Answer

$$\ln y=-4x+\ln 6$$
$$\ln y-\ln 6=-4x,\ln \frac{ y }{ 6 }=-4x,y=6e ^{-4x}$$
$$or~f \left( x \right)=6e ^{-4x}$$

anonymous · Answer

Ah... Ok, those e rules I must remember... Thank you very much for your help.

anonymous · Answer

yw