91Ó°ÊÓ

Involve exponential decay. The radioactive element iodine- 131 has a decay constant of -1.3863 day \(^{-1} .\) Find its half-life.

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{x y}{1+x^{2}}$$

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.3 x-0.2 x^{2}-0.2 x y \\ y^{\prime}=0.2 y-0.1 y^{2}-0.2 x y \end{array}\right.$$

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=\sqrt{x+y}, y(0)=1$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{2}{x y+y}$$

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=\sqrt{x^{2}+y^{2}}, y(0)=4$$

Involve exponential decay. The radioactive element cesium- 137 has a decay constant of -0.023 year \(^{-1} .\) Find its half-life.

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.4 x-0.3 x^{2}-0.1 x y \\ y^{\prime}=0.3 y-0.2 y^{2}-0.1 x y \end{array}\right.$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{\cos ^{2} y}{4 x-3}$$

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.2 x-0.2 x^{2}-0.1 x y \\ y^{\prime}=0.1 y-0.1 y^{2}-0.2 x y \end{array}\right.$$