Consider the system

${x}^{\prime}=\frac{-x}{2};{y}^{\prime}=2y+{x}^{2}$

Show that this system is topologically conjugate to the linear system $D{F}_{(0,0)}$

a) Solve both the linear and nonlinear systems and express your answers as a flows ${\varphi}_{t}^{L}(x,y)$ and ${\varphi}_{t}^{N}(x,y)$ respectively.

b) As I found ${\varphi}_{t}^{L}({x}_{0},{y}_{0})$ = $({x}_{0}{e}^{\frac{-1}{2}t},{y}_{0}{e}^{2t})$ and ${\varphi}_{t}^{N}({x}_{0},{y}_{0})$ = $({x}_{0}{e}^{\frac{-1}{2}t},({y}_{0}+\frac{1}{3}{x}_{0}^{2}){e}^{2t}-\frac{{x}_{0}^{2}}{3}{e}^{-t})$ Do they look right to you?

c) Find the topological conjugacy that maps the flow of the nonlinear system to that of the linear system.

${x}^{\prime}=\frac{-x}{2};{y}^{\prime}=2y+{x}^{2}$

Show that this system is topologically conjugate to the linear system $D{F}_{(0,0)}$

a) Solve both the linear and nonlinear systems and express your answers as a flows ${\varphi}_{t}^{L}(x,y)$ and ${\varphi}_{t}^{N}(x,y)$ respectively.

b) As I found ${\varphi}_{t}^{L}({x}_{0},{y}_{0})$ = $({x}_{0}{e}^{\frac{-1}{2}t},{y}_{0}{e}^{2t})$ and ${\varphi}_{t}^{N}({x}_{0},{y}_{0})$ = $({x}_{0}{e}^{\frac{-1}{2}t},({y}_{0}+\frac{1}{3}{x}_{0}^{2}){e}^{2t}-\frac{{x}_{0}^{2}}{3}{e}^{-t})$ Do they look right to you?

c) Find the topological conjugacy that maps the flow of the nonlinear system to that of the linear system.