AP Calculus BC 7.7 Particular DE Solutions Overview

Review particular solutions by applying an initial condition to determine constants after solving the differential equation.

Covered syllabus points

FUN-7.E—Determine particular solutions to differential equations

[Maximum number: 5]

The depth of seawater at a location can be modeled by the function H that satisfies the differential equation $\frac{d H}{d t}=\frac{1}{2}(H-1) \cos \left(\frac{t}{2}\right)$ , where H(t) is measured in feet and t is measured in hours after noon ( t=0 ). It is known that H(0)=4.

Use separation of variables to find y=H(t), the particular solution to the differential equation $\frac{d H}{d t}=\frac{1}{2}(H-1) \cos \left(\frac{t}{2}\right)$ with initial condition H(0)=4.

Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.

Figure for Question 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables question 1 — AP Calculus BC

Separate variables:
$\frac{dH}{H-1}=\frac{1}{2}\cos\left(\frac{t}{2}\right)dt.$
Integrating gives
$\ln|H-1|=\sin\left(\frac{t}{2}\right)+C.$
Using H(0)=4, $\ln 3=C$ . Since H>1, |H-1|=H-1, so
$\ln(H-1)=\sin\left(\frac{t}{2}\right)+\ln 3.$
Therefore
$H(t)=1+3e^{\sin(t/2)}.$

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Covered syllabus points

7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables question 1