Additional Exercises for Chapter 3

Qualitative Methods for First-Order Differential Equations

1. For each of the following first-order differential equations, determine the number of fixed points. If possible, find the fixed points exactly, Which of the fixed points are stable?
   1. $$\dot x = x^3-7x+6$$
   2. $$ \dot x = 3x^4 -2x^3 - 3 x^2 -1$$
   3. $$\dot x =e^x-x^3$$
2. Find the fixed points of $$\dot x = 1+\cos x.$$ Investigate their stability, and hence describe the behaviour of solutions of the differential equation for all possible initial conditions.
3. Consider the differential equation $$\dot x = x^2-t.$$ By considering the direction field, describe the solution with the initial condition \(x(0)=0\). Find an approximate formula for \(x(t)\) for large \(t\).
4. For each of the following differential equations, give a qualitative description of its behaviour for all possible initial conditions.
   1. $$ \dot x = x^2 - 5x +7 $$
   2. $$ \dot x = -|x| $$
   3. $$ \dot x =\frac{ x(x-t)}{1+t}$$ (hint: it may help to search for the two straight line solutions)

Answers

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