Exploring Solutions for Differential Equations and Simple Harmonic Motion

[wel]

Simple harmonic motion: $y'= -z,~z'= f(y)$the modified explicit equation are$$y'=-z+\frac {1}{2} hf(y)$$$$y'=f(y)+\frac {1}{2} hf_y z$$
and deduce that the coresponding approximate solution lie on the family of curves
$$2F(y)-hf(y)y+z^2=\textrm{constant}$$where $f_y= f(y)$.
What are the curves when verify $f(y)=y$=> for the solution of the system lie on the family of curves, i was thinking$$\frac{d}{dt}[F(y)^2+z^2]= y \frac{dy}{dt} + z \frac{dz}{dt}$$but i am not sure if $f(y)=y$, then the differential equation is $y'' + y =0$, meaning that $y=A \cos x +B \sin x$ and $z=-y'= - A \sin x +B \cos x$.

 

[micromass]

Dear wel,

This thread involves derivatives and diff eq. Thus it belongs in the Calculus & Beyond homework and not precalculus. Please post there in the future.