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Magazine Problem 5
In Problems 1-6 write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+x=\epsilon x^{3} \text { for } \epsilon>0 $$
Problem 5
Without solving explicitly, classify the critical points of the given first- order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d T}{d t}=k\left(T-T_{0}\right) $$
Problem 6
In Problems 1-6 write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+x-\epsilon x|x|=0 \text { for } \epsilon>0 $$
Problem 6
Answer Problems 1-10 without referring back to the text. Fill in the blank or answer true/false. If the Jacobian matrix \(\mathbf{A}=\mathbf{g}^{\prime}\left(\mathbf{X}_{1}\right)\) at a critical point of a plane autonomous system has positive trace and determinant, then the critical point \(\mathbf{X}_{1}\) is unstable.
Problem 6
Without solving explicitly, classify the critical points of the given first- order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ m \frac{d v}{d t}=m g-k v $$
Problem 7
Without solving explicitly, classify the critical points of the given first- order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d x}{d t}=k(\alpha-x)(\beta-x), \quad \alpha>\beta $$
Problem 7
In Problems 7-16 find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x+x y \\ &y^{\prime}=-y-x y \end{aligned} $$
Problem 8
Without solving explicitly, classify the critical points of the given first- order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d x}{d t}=k(\alpha-x)(\beta-x)(\gamma-x), \quad \alpha>\beta>\gamma $$
Problem 8
Answer Problems 1-10 without referring back to the text. Fill in the blank or answer true/false. All solutions to the pendulum equation \(\frac{d^{2} \theta}{d t^{2}}+\frac{g}{l} \sin \theta=0\) are periodic.
Problem 8
In Problems 7-16 find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=y^{2}-x \\ &y^{\prime}=x^{2}-y \end{aligned} $$
