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### Mixing. Suppose a brine containing 0.3 kilogram (…

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Case Western Reserve University
Problem 32

# Analyze the solution $y=\phi(x)$ to the initial value problem$$\frac{d y}{d x}=y^{2}-3 y+2, \quad y(0)=1.5$$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 $\lim _{x \rightarrow \infty} \phi(x)$(b) Use Euler's method with a step size of 0.1 to find an approximation of $\phi(1)$.(c) Find a formula for $\phi(x)$ and graph $\phi(x)$ on the direction field from part (a).(d) What is the exact value of $\phi(1) ?$ Compare with your approximation in part (b).(e) Using the exact solution obtained in part (c), determine $\lim _{x \rightarrow \infty} \phi(x)$ and compare with your guess in part (a).

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