7.
7 Separation of Variables (Particular Solutions)
Calculus
Practice
For each differential equation, find the solution that passes through the given initial condition.
1. 𝑦 cos 𝑥 and 𝑦 4 when 𝑥 0 2. if 𝑦 0 2
3. 𝑦 sin 𝑥 if 𝑦 2 4. 8 𝑦 and 𝑦 6 when 𝑥 0
5. 𝑦 5 𝑥 2 when 𝑓 0 1 6. if 𝑦 0 4
7. Find the particular solution to 𝑒 when 𝑓 0 2. Sketch the
graph of this particular solution on the slope field provided.
8. Find the particular solution to 𝑥𝑦 if 𝑦 1 when 𝑥 0. Sketch
the graph of this particular solution on the slope field provided.
7.7 Separation of Variables (Particular Solutions) Test Prep
9. Consider the differential equation 𝑒 4𝑥 1 . Let 𝑦 𝑓 𝑥 be the particular solution to the differential
equation that passes through 2,0 .
(a) Write an equation for the line tangent to the graph of 𝑓 at the point 2,0 . Use the tangent line to
approximate 𝑓 2.2 .
(b) Find 𝑦 𝑓 𝑥 , the particular solution to the differential equation that passes through 2,0 .
10. One of Mr. Kelly’s calculus students attempted to solve the differential equation 2𝑥𝑦 with initial
condition 𝑦 3 when 𝑥 0. In which step, if any, does an error first appear?
Step 1: 𝑑𝑦 2𝑥 𝑑𝑥
Step 2: ln|𝑦| 𝑥 𝐶
Step 3: |𝑦| 𝑒 𝐶
Step 4: Since 𝑦 3 when 𝑥 0, 3 𝑒 𝐶.
Step 5: 𝑦 𝑒 2
(A) Step 2 (B) Step 3 (C) Step 4 (D) Step 5 (E) There is no
error in the solution.
11. Consider the differential equation 6 2𝑦. Let 𝑦 𝑓 𝑥 be the particular solution to the differential
equation with the initial condition 𝑓 0 4.
a. Write an equation for the line tangent to the graph of 𝑦 𝑓 𝑥 at 𝑥 0. Use the tangent line to
approximate 𝑓 0.6 .
b. Find the value of at the point 0, 4 . Is the graph of 𝑦 𝑓 𝑥 concave up or concave down at the point
0, 4 ? Give a reason for your answer.
c. Find 𝑦 𝑓 𝑥 , the particular solution to the differential equation with the initial condition 𝑓 0 4.
d. For the particular solution 𝑦 𝑓 𝑥 found in part c, find lim 𝑓 𝑥 .
→
12. The graph of the derivative of 𝑓, 𝑓 , is shown to the right. The y
domain of 𝑓 is the set of all 𝑥 such that 4 𝑥 0.
𝑓 𝑥
Given that 𝑓 2 0, find the solution 𝑓 𝑥 .
x
The function exists at 𝑥 2, so there
should be an “or equal to” on one of
the inequalities.
13. Mr. Bean’s favorite addiction (rhymes with
Poctor Depper) is put into a cylindrical container with radius 3 inches, as shown in 3 in
the figure above. Let ℎ be the depth of the soda in the container, measured in
inches, where ℎ is a function of time 𝑡, measured in minutes. The volume 𝑉 of soda
in the container is changing at the rate of √ℎ cubic inches per minute throughout
the morning. Given that ℎ 9 at the start of 1st period (𝑡 0), solve the differential
equation for ℎ as a function of 𝑡. (The volume 𝑉 of a cylinder with radius 𝑟 and
ℎ in
height ℎ is 𝑉 𝜋𝑟 ℎ.)
14. The rate at which a baby koala bear gains weight is proportional to the difference between its adult weight and
its current weight. At time 𝑡 0, when the bear is first weighed, its weight is 2 pounds. If 𝐵 𝑡 is the weight
of the bear, in pounds, at time 𝑡 days after it is first weighed, then 20 𝐵 . Find 𝑦 𝐵 𝑡 , the
particular solution to the differential equation.
