Separation of Variables: Day 2 Section 6.3 Calculus AP/Dual, Revised ©2018 viet.dang@humbleisd.net 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Review 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Example 1 Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝟏−𝒚 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Example 1 Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝟏−𝒚 8/31/2019 10:17 PM §6.3: Separation of Variables

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. 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Example 2 Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Example 2 Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Example 2 Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Example 2 Solve for the differential equation, 𝒚 ′ 𝟏+ 𝒙 𝟐 −𝟐𝒙𝒚=𝟎 8/31/2019 10:17 PM §6.3: Separation of Variables

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 ? 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Example 3 Solve for the differential equation, 𝒅𝒚 𝒅𝒙 =𝒙 𝒚 𝟐 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Your Turn Solve for the differential equation, 𝒅𝒚 𝒅𝒙 = 𝒙 𝟐 𝒚+𝟏 8/31/2019 10:17 PM §6.3: Separation of Variables

§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 8/31/2019 10:17 PM §6.3: Separation of Variables

§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 8/31/2019 10:17 PM §6.3: Separation of Variables

§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 8/31/2019 10:17 PM §6.3: Separation of Variables

§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 8/31/2019 10:17 PM §6.3: Separation of Variables

§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 8/31/2019 10:17 PM §6.3: Separation of Variables

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 2010. Find the particular solution 𝑾=𝑾 𝒕 to the differential equation 𝒅𝑾 𝒅𝒕 = 𝟏 𝟐𝟓 𝑾−𝟑𝟎𝟎 with initial condition 𝑾 𝟎 =𝟏𝟒𝟎𝟎. 8/31/2019 10:17 PM §6.3: Separation of Variables

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. 8/31/2019 10:17 PM §6.3: Separation of Variables

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. 8/31/2019 10:17 PM §6.3: Separation of Variables

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. 8/31/2019 10:17 PM §6.3: Separation of Variables

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. 8/31/2019 10:17 PM §6.3: Separation of Variables

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) 8/31/2019 10:17 PM §6.3: Separation of Variables

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) 8/31/2019 10:17 PM §6.3: Separation of Variables

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) 8/31/2019 10:17 PM §6.3: Separation of Variables

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. 8/31/2019 10:17 PM §6.3: Separation of Variables

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. 8/31/2019 10:17 PM §6.3: Separation of Variables

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 8/31/2019 10:17 PM §6.3: Separation of Variables

§6.3: Separation of Variables Assignment Worksheet 8/31/2019 10:17 PM §6.3: Separation of Variables