Chapter 2.2, Problem 32E

Analyze the solution $y=\varphi \left(x\right)$ to the initial value problem

$\frac{dy}{dx}=y^2-3y+2$, $y\left(0\right)=1.5$

using approximation methods and then compare with its exact form as follows.

a. Sketch the direction field of the differential equation and use it to guess the value of $\underset{x\to \infty }{\lim }\varphi \left(x\right)$.

b. Use Euler’s method with a step size of $0.1$ to find an approximation of $\varphi \left(1\right)$.

c. Find a formula for $\varphi \left(x\right)$ and graph $\varphi \left(x\right)$ on the direction field from part (a).

d. What is the exact value of $\varphi \left(1\right)$? Compare with your approximation in part (b).

e. Using the exact solution obtained in part (c), determine $\underset{x\to \infty }{\lim }\varphi \left(x\right)$, and compare with your guess in part (a).