Question

any body can help me with solving following equition: d^2(a)/ds -(d(a)/ds)*sina +cosa +k =0 actually it is like this y"-y'*sin(x)+cos(x)+k=0 where k is fix number

Answer 1

actually it is like this y"-y'*sin(x)+cos(x)+k=0 where k is fix number

Answer 2

First consider the case $$z'-z\sin x=-\cos x-k$$Notice it is a *linear differential equation* in \(z=y'\). Observe we can multiply by an integration factor \(\mu=e^{-\int\sin x\,dx}=e^{\cos x}\) to get:$$e^{\cos x}z'-e^{\cos x}z\sin x=-e^{\cos x}\cos x-e^{\cos x}k\\(e^{\cos x}z)'=-e^{\cos x}\cos x-e^{\cos x}k\\e^{\cos x}z=-\int e^{\cos x}(\cos x+k)\,dx\\z=-e^{-\cos x}\int e^{\cos x}(\cos x+k)\,dx\\y=-\int e^{-\cos x}\int e^{\cos x}(\cos x+k)\,dx\,dx$$I'm not sure if there's a nice form for this.

Answer 3

thank you I will work on that to see with this solution if I would find acceptable practical result or not and I would come back to you actually it is driven from practical engineering problem and it demonstrating centenary shape of pipeline during laying variable here is angle

Answer 4

could you please help we with case illustrate in enclosed\[\alpha*d (\theta)^{2}/d s^{2}-v*\sin \theta+h*\cos \theta +k =0\] file