5.1 Separation of Variables (Day 1)

Introduction to Integrating Differential Equations

• New unit on applications of integrals starts with integrating differential equations with two variables.

• Method: Separation of Variables.

Steps for Separation of Variables

Step 1: Rearranging the Equation

• Move all y's to the left side and all x's to the right side.

• Structure: y's and dy's on the left; x's and dx's on the right.

Step 2: Integrate Both Sides

• Integrate left side with respect to y and right side with respect to x.

Step 3: Solve for y

• Isolate y by performing necessary algebraic manipulations.

Example Problem 1: Integrating y and x

• Start with ( y \frac{dy}{dx} = 2x ).

• Multiplying both sides by dx: ( y \frac{dy}{dx} dx = 2x dx ).

• Resulting in ( y dy = 2x dx ).

• Integrate both sides:

  • ( \int y dy = \int 2x dx )

    • Left side: ( \frac{y^2}{2} + C_1 )

    • Right side: ( x^2 + C_2 )

• Combine constants to get ( \frac{y^2}{2} = x^2 + C ).

• Final form: ( y = \sqrt{2x^2 + C} ).

Example Problem 2: Revising the Equation

• Rearranging: ( dy = y^2 dx )

• Dividing by y^2: ( \frac{1}{y^2} dy = dx ).

• Integrating gives: ( -\frac{1}{y} = x + C ).

• Isolating y leads to: ( y = -\frac{1}{x + C} ).

Example Problem 3: Incorporating Trigonometric Functions

• Equation: ( 3y^2 \frac{dy}{dx} = x + \sin(x) ).

• Rearranging: ( 3y^2 dy = (x + \sin(x)) dx ).

• When integrated:

  • Left: ( y^3 + C_1 )

  • Right: ( \frac{1}{2}x^2 - \cos(x) + C_2 )

• Isolating y results in: ( y = \sqrt[3]{\frac{1}{2}x^2 - \cos(x) + C} ).

Unique Case: Using ln Function

• Given: ( dy = xy dx ) leads to ( \frac{1}{y} dy = x dx ).

• Integration yields: ( \ln(y) = \frac{1}{2}x^2 + C ).

• To isolate y, exponentiate both sides:

  • ( y = e^{\frac{1}{2}x^2 + C} ).

  • Result: ( y = C e^{\frac{1}{2}x^2} ) where C is a constant.