Unique Solution differential equation system/ linear limited

Question

I have the following differential equation system:

$$\begin{align} x'(t)&=\sqrt{1+x^2} +y^3 \sin x -x^7\\ y'(t)&=x(1-y^2 \sin x) \end{align}$$ with $ x(0)=x_0, \ y(0)=y_0$

I have to show, that $ \forall t \geq 0 $ the System has a unique solution:

I have to show, that the right hand side is linear limited: $|f(t,x,y)|\leq a(t) + b(t)|(x,y)| $ Can somebody explain me, how I manage to do that?

Answer 1

The solutions are indeed bounded, as the graphic below demonstrates. To get a theoretical proof of that, consider the radius square $V(x,y)=x^2+y^2$ so that along solutions you get for $v(t)=V(x(t),y(t))$ \begin{align} v'(t)=\frac{d}{dt}(x(t)^2+y(t)^2)=2(xx'+yy') &=2\left(x\sqrt{1+x^2}-x^8+xy\right)\\ &\le (1+2x^2)+(x^2+y^2)\le 1+3v(t), \end{align} using $2ab\le a^2+b^2$ twice. This differential inequality provides not exactly the boundedness of the solutions, but an exponential bound for the growth of solutions, $$ v(t)\le(\tfrac13+v(0))e^{3t}-\tfrac13. $$ As the solutions can thus not diverge to infinity in finite time, they are defined for all times.

def eqn(x,y): return (1+x**2)**0.5+y**3*np.sin(x)-x**7, x*(1-y**2*np.sin(x))

# generate streamplot
r = 5
X, Y = np.meshgrid(np.arange(-r, r+.01, 0.1), np.arange(-r, r+.05, 0.1))

dotX, dotY = eqn(X,Y)

plt.streamplot(X, Y, dotX, dotY, density=2.5)
plt.xlim(-r,r); plt.ylim(-r,r);
plt.grid(); plt.show()