# MATH 308 Homework 3

```Texas A&amp;M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2010
MATH 308 Homework 3
3.1. Sketch the graph of f (y) versus y, determine the critical (equilibrium) points, and
classify each one as asymptotically stable or unstable.
dy/dt = ay + by 2 ,
a &gt; 0,
b &gt; 0,
−∞ &lt; y0 &lt; ∞.
3.2. Sketch the graph of f (y) versus y, determine the critical (equilibrium) points, and
classify each one as asymptotically stable, unstable, or semistable.
dy/dt = y 2 (y 2 − 1).
3.3. Determine whether the equation is exact. If it is exact, then solve the equation.
dy/dx = −
4x + 2y
.
2x + 3y
3.4. Solve the initial value problem
(2x − y)dx + (2y − x)dy = 0,
y(1) = 3.
3.5. Find the value of b for which the given equation is exact, and then solve it using
that value of b.
(ye2xy + x)dx + bxe2xy dy = 0.
3.6. Show that the given equation is not exact but becomes exact when multiplied by
the given integrating factor. Solve the equation.
x2 y 3 + x(1 + y 2 )y 0 = 0,
&micro;(x, y) = 1/xy 3 .
```