AP Calculus BC Fun 7 E Determine Particular Solutions to Differential Equations Questions

Practice AP Calculus BC questions on using an initial condition to determine a particular differential-equation solution.

[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.

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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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AP Calculus BC Fun 7 E Determine Particular Solutions to Differential Equations Questions question 1