Problem 1
Analyze the local stability of each of the following nonlinear systems: (a) \(x^{\prime}=e^{x}-1\) \(y^{\prime}=y e^{x}\) (b) \(x^{\prime}=x+2 y\) \(y^{\prime}=x^{2}+y\) (c) \(x^{\prime}=1-e^{y}\) \(y^{\prime}=5 x-y\) (d) \(x^{\prime}=x^{3}+3 x^{2} y+y\) \(y^{\prime}=x\left(1+y^{2}\right)\)
Problem 4
Solve the following two differential-equation systems: \((a) x^{\prime}(t) \quad-x(t)-12 y(t)=-60\) \\[ y^{\prime}(t)+x(t)+6 y(t)=36 \quad[\text { with } x(0)=13 \text { and } y(0)=4] \\] (b) \(x^{\prime}(t) \quad-2 x(t)+3 y(t)=10\) \\[ y^{\prime}(t)-x(t)+2 y(t)=9 \quad[\text { with } x(0)=8 \text { and } y(0)=5] \\]
Problem 4
The following two systems both possess zero-valued Jacobians. Construct a phase diagram for each, and deduce the locations of all the equilibrium's that exist: (a) \(x^{\prime}=x+y\) \(y^{\prime}=-x-y\) (b) \(x^{\prime}=0\) \(y^{\prime}=0\)
