# Decidable Theories I by Dirk Siefkes, Gert H. Müller

By Dirk Siefkes, Gert H. Müller

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And that helps you come to this solution: y = t 2 + 32 t And there you have it. You can see a direction field for the many general solutions to this differential equation in Figure 2-3. 33 34 Part I: Focusing on First Order Differential Equations 5 4 3 y 2 1 0 –1 Figure 2-3: The direction field of a more advanced solution. –2 –3 –4 –5 –2 –1 0 1 x 2 You can see this function graphed in Figure 2-4. 8 7 y 6 5 4 Figure 2-4: The graph of 3 a more advanced solution. 2 –2 –1 0 x 1 2 Chapter 2: Looking at Linear First Order Differential Equations Determining Whether a Solution for a Linear First Order Equation Exists I show you how to deal with different kinds of linear first order differential equations earlier in this chapter, but the fact remains that not all linear differential equations actually do have a solution.

On the other hand, a solution to this differential equation is still guaranteed because f(x, y) is continuous. However, it doesn’t guarantee the uniqueness of that solution. Chapter 3 Sorting Out Separable First Order Differential Equations In This Chapter ᮣ Figuring out the fundamentals of separable differential equations ᮣ Applying separable differential equations to real life ᮣ Advancing with partial fractions S ome rocket scientists call you, the Consulting Differential Equation Expert, into their headquarters.

I can see that,” you say. ” “Separable? ” they ask. “Separable means that you can recast the equation like this, where x is on one side and y is on the other,” you say while showing them the following equation on your clipboard: (2 – y2) dy = x2 dx “You can integrate the equation with respect to y on one side, and x on the other,” you say. “We never thought of that. ” 42 Part I: Focusing on First Order Differential Equations That’s what this chapter covers: separable first order differential equations.