i've been trying to solve 1st order differential equations with dsolve, euler andy runge kutta methods and compare them on graphics. for simplification, dy means dy/dx: first equation is y' = exp(-2*x^2)-4*x*y with initial condition y(0) = -4.3 which i have no problem with. it plots as follows:
but using the same code, another diff. eq. plots with a huge error after the critical point. it is y' = (1-y^2)^.5 with initial condition y(0) = 1/sqrt(2)
here is my code: http://freetexthost.com/glmzxekwfd any help appreciated.
The differential equation $$y'(x)=\sqrt{1-y^2(x)},\quad y(0)=1/\sqrt2,$$ is solved by $$y(x)=\left\{\begin{array}{ccc}\sin(x+\pi/4)&\text{if}&0\leqslant x\leqslant\pi/4\\1&\text{if}&x\gt\pi/4\end{array}\right.$$ The "analytik" curve in your figure seems to plot on the interval $(0,2)$ the function $z$ defined by $$z(x)=\sin(x+\pi/4),$$ which does not solve the differential equation on the interval $(\pi/4,2)$, since there, the graph of $z$ has a negative slope although the differential equation indicates that every solution $y$ is such that $y'(x)\geqslant0$ for every $x$. The function $z$ solves on the whole real line the differential equation $$(z'(x))^2=1-z^2(x),\quad z(0)=1/\sqrt2.$$ Another problem arises from the values greater than $1$ in the "euler" and "runge-kutta" plots. You are probably facing various bugs in the solvers you are using.