Finding Particular Solutions Using Initial Conditions and Separation of Variables

#7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

Welcome to topic 7.7, where you’ll learn about how to find unique solutions to differential equations. You’ve almost made it to the end of the unit! 🌟

In this topic, we’ll learn about the difference between a general solution and a particular solution using separation of variables. To review separation of variables, please look at 7.6 linked here!

#🤔 Difference Between General and Particular Solutions

A general solution to a differential equation consists of a constant in the equation. Therefore, when you change the constant, you can get multiple equations when plotting, as seen in the picture below on the left side below. ⬅️ You get a general solution when you don’t have any initial conditions that are given by the problem. When plotting the general solution, you will also get a graph that looks similar to a slope field. For references on slope fields, please look at 7.3 and 7.4!

However, when given an initial condition in the problem, you can solve for a specific equation, which leaves you with one equation on your graph, like on the right side below. ➡️

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A visual representation of general solutions vs. particular solutions with slope fields

Image Courtesy of Higher Math Notes

Essentially…

• 🎩 A general solution to a differential equation is a family of functions that satisfies the equation. There are infinitely many functions that could do so!
• 🎯 A particular solution is a unique solution that passes through a specific point, and we can calculate it when given initial conditions.

#🧠 Particular Solution Function

When this topic shows up on free-response questions and you’re asked to write an expression, you must include the initial condition. Here is a general form that you can use:

$$F(x)=y_0+\int_a^x f(t)dt$$

This is a particular solution to the differential equation $\frac{dy}{dx}=f...