Question

given dy/dx=e^ysin(x),y(0)=-1 solve this differential equation analytically

Answer 1

pls write the question properly.

Answer 2

\[e^{-y}dy=\sin(x)dx \rightarrow \int\limits\limits_{}_{}e^{-y}dy= \int\limits\limits_{}_{}\sin(x)dx \rightarrow e^{-y}=\cos(x)+C\]\[y=-Ln[\cos(x)+C]\rightarrow y(0)=-Ln[1+C]=-1\rightarrow 1+C=e \rightarrow C=e-1\]\[y(x)=-Ln[\cos(x)+e-1]\]Test it:
\[\frac{ dy }{ dx }=\frac{ \sin(x) }{ \cos(x)+e-1 }=e^{-Ln[\cos(x)+e-1]}\sin(x)=e^ysin(x)\]Also, \[y(0)=-Ln[\cos(0)+e-1]=-Ln(e)=-1\]

Answer 3

separate variables and integrate:
\(\int e^{-y}dy=\int sinx dx +C\)
\(-e^{-y} = -cosx +C\)
\(e^{-y}=cosx-C\)
now use initial condition:
\(e^{-1}=cos(0) -C\)
so \(C=1-1/e=e-1\)
and answer is:
\(e^{-y}=cosx-e+1\)

Answer 4

@myko Review again initial conditions:\[e^1\neq \cos(0)-e+1=2-e\]

Answer 5

no

Answer 6

You made a mistake with C. C=e-1 and not 1-e

Answer 7

you have integration mistake:
\(\int e^{-y} \neq e^{-y} \)
\(\int e^{-y}=-e^{-y}\)

Answer 8

Absolutely no mistake at all in my integral. Left side has negative sign but integral of sin(x) is -cos(x), that is why I have eliminated minus sign on both sides

Do not review my calculations, your solution does not fulfills initial conditions, thats all

Answer 9

my solution does, btw

Answer 10

http://www.wolframalpha.com/input/?i=solve+for+y+dy%2Fdx%3De%5Eysin%28x%29+y%280%29%3D1

Answer 11

Yes, nice program but you need to input the right data. The problem states y(0)=-1 and you have input y(0)=1, Sir

Answer 12

oh, my bad I didn't see it was y(0)=-1. I used y(0)=1

Answer 13

Now input y(0)=-1 and compare to my solution above

Answer 14

guys the questions is solve this diffential equation analytically showing all stepsnand express the solution in the form y = f(x) GIVEN: dy/dx=e^ysin(x), y(0)=-1

Answer 15

@Mokete seems like you have not read the posts above, where the whole process is described

Answer 16

@carlosGP ooh ya i saw them. thanx

Answer 17

using a step size h=0.5,compute the numerical solutions of the given differential equation in the interval 0<= t <= 1 using the 4th order Runge-kutta method
[GIVEN: dy/dx=e^y sin(x), y(0)=-1