91Ó°ÊÓ

In Problems first solve the equation \(f(x)=0\) to find the critical points of the given autonomous differential equation \(d x / d t=f(x) .\) Then analyze the sign of \(f(x)\) to determine whether each crifical point is stable or unstable, and construct the corresponding phase diagrant for the differential equarion. Next, solve the differential equarion explicitly for \(x(t)\) in terms of t. Finally use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point. $$ \frac{d x}{d t}=(x-2)^{2} $$

An initial value problem and its exact solution \(y(x)\) are given. Apply Euler's method twice to approximate to this solution on the interval \(\left[0, \frac{1}{2}\right]\), first with step size \(h=0.25\), then with step size \(h=0.1 .\) Compare the threedecimal-place values of the two approximations at \(x=\frac{1}{2}\) with the value \(y\left(\frac{1}{2}\right)\) of the actual solution. $$ y^{\prime}=-3 x^{2} y, v(0)=3 ; y(x)=3 e^{-x^{3}} $$

Find the exact solution of the given initial value problem. Then apply Euler's method twice to approximate (to four decimal places) this solution on the given interval, first with step size \(h=0.01\), then with step size \(h=0.005 .\) Make a table showing the approximate values and the actual value, together with the percentage error in the more accurate approximation, for \(x\) an integral multiple of 0.2. Throughout, primes denote derivatives with respect to \(x\). $$ y^{2} y^{\prime}=2 x^{5}, y(2)=3 ; 2 \leqq x \leqq 3 $$

Use Euler's method with step sizes \(h=0.1,0.02,0.004\), and \(0.0008\) to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. Print the results in tabular form with appropriate headings to make it easy to gauge the effect of varying the step size h. Throughout, primes denote derivatives with respect to \(x\). $$ y^{\prime}=\ln y, y(1)=2 ; 1 \leqq x \leqq 2 $$

Apply Euler's method with successively smaller step sizes on the interval \([0,2]\) to verify empirically that the solution of the initial value problem $$ \frac{d y}{d x}=x^{2}+y^{2}, \quad y(0)=0 $$ has a vertical asymptote near \(x=2.003147 .\) (Contrast this with Example 2 , in which \(y(0)=1\).)