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Magazine Problem 1
a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$\frac{d y}{d x}=(y+2)(y-3)$$
Problem 1
Solve the differential equations $$x \frac{d y}{d x}+y=e^{x}, \quad x>0$$
Problem 2
a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$\frac{d y}{d x}=y^{2}-4$$
Problem 2
Solve the differential equations $$e^{x} \frac{d y}{d x}+2 e^{x} y=1$$
Problem 2
For the system ( \(2 a\) ) and ( 2 b), show that any trajectory starting on the unit circle \(x^{2}+y^{2}=1\) will traverse the unit circle in a periodic solution. First introduce polar coordinates and rewrite the system as \(d r / d t=r\left(1-r^{2}\right)\) and \(-d \theta / d t=-1\).
Problem 3
Solve the differential equations $$x y^{\prime}+3 y=\frac{\sin x}{x^{2}}, \quad x>0$$
Problem 3
a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$\frac{d y}{d x}=y^{3}-y$$
Problem 4
Solve the differential equations $$y^{\prime}+(\tan x) y=\cos ^{2} x, \quad-\pi / 2< x < \pi / 2$$
Problem 4
a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$\frac{d y}{d x}=y^{2}-2 y$$
Problem 5
a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) c. Sketch several solution curves. $$y^{\prime}=\sqrt{y}, \quad y>0$$
