1. Separation of Variables: Day 2
Section 6.3
Calculus AP/Dual, Revised ©2018
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§6.3: Separation of Variables

2. §6.3: Separation of Variables
Review
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§6.3: Separation of Variables

3. §6.3: Separation of Variables
Example 1
Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝟏−𝒚
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§6.3: Separation of Variables

4. §6.3: Separation of Variables
Example 1
Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝟏−𝒚
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§6.3: Separation of Variables

5. Example 1 Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝟏−𝒚
𝒆 𝒄 will always give us a constant. Therefore, we put it in front.
Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝟏−𝒚
Technically, 𝑪 is a constant that is either positive or negative. Therefore, the absolute value is not needed at all.
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§6.3: Separation of Variables

6. §6.3: Separation of Variables
Example 2
Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎
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§6.3: Separation of Variables

7. §6.3: Separation of Variables
Example 2
Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎
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§6.3: Separation of Variables

8. §6.3: Separation of Variables
Example 2
Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎
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§6.3: Separation of Variables

9. §6.3: Separation of Variables
Example 2
Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎
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§6.3: Separation of Variables

10. What is the integral of 𝒅𝒚 𝒚 𝟐 ? §6.3: Separation of Variables
Example 3
Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝒚 𝟐
What is the integral of 𝒅𝒚 𝒚 𝟐 ?
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§6.3: Separation of Variables

11. §6.3: Separation of Variables
Example 3
Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝒚 𝟐
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§6.3: Separation of Variables

12. §6.3: Separation of Variables
Your Turn
Solve for the differential equation, 𝒅𝒚 𝒅𝒙 = 𝒙 𝟐 𝒚+𝟏
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§6.3: Separation of Variables

13. §6.3: Separation of Variables
Example 4
Given the initial condition 𝒚 𝟎 =𝟏, find the particular solution of the equation, 𝒙𝒚 𝒅𝒙+ 𝒆 − 𝒙 𝟐 𝒚𝟐−𝟏 𝒅𝒚=𝟎
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§6.3: Separation of Variables

14. §6.3: Separation of Variables
Example 4
Given the initial condition 𝒚 𝟎 =𝟏, find the particular solution of the equation, 𝒙𝒚 𝒅𝒙+ 𝒆 − 𝒙 𝟐 𝒚𝟐−𝟏 𝒅𝒚=𝟎
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§6.3: Separation of Variables

15. §6.3: Separation of Variables
Example 4
Given the initial condition 𝒚 𝟎 =𝟏, find the particular solution of the equation, 𝒙𝒚 𝒅𝒙+ 𝒆 − 𝒙 𝟐 𝒚𝟐−𝟏 𝒅𝒚=𝟎
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§6.3: Separation of Variables

16. §6.3: Separation of Variables
Example 5
Given the initial condition 𝟎, 𝟖 , find the particular equation of the equation, 𝒅𝒚 𝒅𝒙 =𝟒 𝒙 𝟑 𝒚
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§6.3: Separation of Variables

17. §6.3: Separation of Variables
Your Turn
Given the initial condition 𝟏, 𝟑 , find the particular equation of the equation, 𝒅𝒚 𝒅𝒙 = 𝒚 𝒙 𝟐
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§6.3: Separation of Variables

18. AP Free Response Question 1 (non-calculator)
At the beginning of 2010, a landfill contained 𝟏𝟒𝟎𝟎 tons of solid waste. The increasing function 𝑾 models the total amount of solid waste stored at the landfill. Planners estimate that 𝑾 will satisfy the differential equation 𝒅𝑾 𝒅𝒕 = 𝟏 𝟐𝟓 𝑾−𝟑𝟎𝟎 for the next 20 years. 𝑾 is measured in tons, and 𝒕 is measured in years from the start of Find the particular solution 𝑾=𝑾 𝒕 to the differential equation 𝒅𝑾 𝒅𝒕 = 𝟏 𝟐𝟓 𝑾−𝟑𝟎𝟎 with initial condition 𝑾 𝟎 =𝟏𝟒𝟎𝟎.
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§6.3: Separation of Variables

19. AP Free Response Question 1 (non-calculator)
Find the particular solution 𝑾=𝑾 𝒕 to the differential equation 𝒅𝑾 𝒅𝒕 = 𝟏 𝟐𝟓 𝑾−𝟑𝟎𝟎 𝑾−𝟑𝟎𝟎 with initial condition 𝑾 𝟎 =𝟏𝟒𝟎𝟎.
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§6.3: Separation of Variables

20. AP Free Response Question 1 (non-calculator)
Find the particular solution 𝑾=𝑾 𝒕 to the differential equation 𝒅𝑾 𝒅𝒕 = 𝟏 𝟐𝟓 𝑾−𝟑𝟎𝟎 𝑾−𝟑𝟎𝟎 with initial condition 𝑾 𝟎 =𝟏𝟒𝟎𝟎.
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§6.3: Separation of Variables

21. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐.
Sketch the slope filed with the given differential equation at the twelve points indicated.
Write an equation for the tangent to the graph of 𝒇 at 𝒙=−𝟏. Then, solve for 𝒇(𝟏.𝟐).
Find the solution of 𝒚=𝒇 𝒙 to the given differential equation with the initial condition 𝒇 −𝟏 =𝟐.
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§6.3: Separation of Variables

22. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐.
Sketch the slope filed with the given differential equation at the twelve points indicated.
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§6.3: Separation of Variables

23. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐.
b) Write an equation for the tangent to the graph of 𝒇 at 𝒙=−𝟏. Then, solve for 𝒇(𝟏.𝟐)
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§6.3: Separation of Variables

24. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐.
b) Write an equation for the tangent to the graph of 𝒇 at 𝒙=−𝟏. Then, solve for 𝒇(𝟏.𝟐)
8/31/ :17 PM
§6.3: Separation of Variables

25. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐.
b) Write an equation for the tangent to the graph of 𝒇 at 𝒙=−𝟏. Then, solve for 𝒇(𝟏.𝟐)
8/31/ :17 PM
§6.3: Separation of Variables

26. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐. c) Find the solution of 𝒚=𝒇 𝒙 to the given differential equation with the initial condition 𝒇 −𝟏 =𝟐.
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§6.3: Separation of Variables

27. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐. c) Find the solution of 𝒚=𝒇 𝒙 to the given differential equation with the initial condition 𝒇 −𝟏 =𝟐.
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§6.3: Separation of Variables

28. AP Free Response Question 2 (non-calculator)
Consider the differentiable equation, 𝒅𝒚 𝒅𝒙 = −𝒙 𝒚 𝟐 𝟐 . Let 𝒚=𝒇 𝒙 be the particular solution to this differential equation with the initial condition 𝒇 −𝟏 =𝟐. c) Find the solution of 𝒚=𝒇 𝒙 to the given differential equation with the initial condition 𝒇 −𝟏 =𝟐.
Raise both sides by exponent of −𝟏 to move the 𝒚 to the front
Divide by −𝟏
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§6.3: Separation of Variables

29. §6.3: Separation of Variables
Assignment
Worksheet
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§6.3: Separation of Variables