This document discusses using integration to solve differential equations that model position, velocity, and acceleration problems. It provides examples of finding general solutions that satisfy initial conditions and particular solutions for specific problems, including: 1) Finding the particular solution that satisfies a given initial condition like passing through a known point. 2) Modeling the motion of a ball thrown upward and using integration to find the position function to determine when the ball hits the ground. 3) The process can also be used to analyze linear motion problems resulting from other forces besides gravity.

1. Particular solutions vs general solutions Position, velocity, acceleration problems

2. You should realize that an antiderivative of a product or a quotient CANNOT be handled in parts! Just like we’ve seen with derivatives of products and quotients, we can’t derive things “in place”. That also applies to products or quotients inside integrands. They have to be rewritten to be able to use an existing rule. Two other warnings: Always include the “+ C” for the indefinite integrals (antiderivatives) and ALWAYS make sure the dx or d-whatever is in the integrand. I look at those very critically!

3. Initial Conditions and Particular Solutions You have seen that we get a family of solutions when we solve y = ∫ f(x) dx since we always add a constant of integration, + C onto the end. This means that the graphs of any two members of the family are VERTICAL translations of each other. y = ∫ (3x 2 – 1)dx y = x 3 – x + C Through (2, 4)

4. y = x 3 – x + C is the general solution, and we want the particular solution that goes through the point (2, 4) 4 = 2 3 – 2 + C, so C = -2 The particular solution is y = x 3 – x - 2

5. Ex 7, p.253 Finding a particular solution Find the general solution of F’(x) = 1/x 2 . Then find the particular solution that goes through the point (1, 2) Solution: Find F(x) = ∫ Rewrite F’(x) Find general solution Plug in (1, 2) to find C. Write particular solution GO!

8. We can do the same kind of process to analyze linear motion problems (vertical or horizontal) that is the result of an acceleration due to some other force besides gravity. 4.1b p. 255/ ( 35-45 odd), 49, 51, 55-61 odd, 67-74 all, 77, 78