I want to analyze the projectile motion when throwing a ball under an angle with air friction.
Now Ive got the thow ODEs:
$$y''(t)-\frac{b}{m}y'(t)=-g$$ and $$x''(t)-\frac{b}{m}x'(t)=0$$
I have $y(t)=\frac{c_1}{k}e^{kt}+\frac{gt}{k}+c_2$ and $x(t)=\frac{c_3}{k}e^{kt}+c_4$ as solutions.
But after determining the $c_i$ and solving $x(t)$ for $t$ I become a function $y(x)$ which is increasing. My problem is now that $y(x)$ clearly doesnt describe the projectile motion of a thrown object. Why is that? Where is my mistake?
The first equation comes from Newton's second law, $F=ma$. You should write it in the form
$$my''(t)=-g-by'(t)$$
where the left side is $ma$ and the right side is the force. You have implicitly defined negative to mean down (by the sign on your $g$, and thus up is positive. Well, when the ball is moving up, the air resistance pushes it down, not up. And when the ball is moving down, the air resistance is pushing it up. Your formula showed the opposite. Your first equation should read
$$y''(t)+\frac{b}{m}y'(t)=-g$$
You might also reconsider your work with $x(t)$, as it might make more sense to you to define the motion to be in the positive direction. Then you expect $x''(t)<0$, so you might wish to write $x''(t)=-\frac{b}{m}x'(t)$, but this is not necessary as your equation will still work. Your formatting might be counter-intuitive to yourself, but is not inherently incorrect.