Integral Calculus - Examples on Variable Separable Method

  • Variable separable is a method used to solve ordinary differential equations of the form dy/dx = f(x) * g(y)

  • The general steps to solve such equations are as follows:

    1. Separate the variables by moving all terms involving y to one side and all terms involving x to the other side.

    2. Integrate both sides of the equation.

    3. Solve for y and simplify if necessary.

    Now, let’s work through some examples to understand this method better.

Example 1

Solve the differential equation: dy/dx = x * y Solution: Step 1: Separate variables dy/y = x * dx Step 2: Integrate ∫ (1/y) dy = ∫ x dx Step 3: Solve for y ln|y| = (1/2) x^2 + C (where C is the constant of integration) |y| = e^((1/2) x^2 + C) y = ± e^(C) * e^((1/2) x^2)

Example 2

Solve the differential equation: dy/dx = 3x^2 - 2y Solution: Step 1: Separate variables dy + 2y = 3x^2 dx Step 2: Integrate ∫ (dy + 2y) = ∫ (3x^2) dx Step 3: Solve for y y + y^2 = x^3 + C

Example 3

Solve the differential equation: dy/dx = y^2 / x^2 Solution: Step 1: Separate variables x^2 dy = y^2 dx Step 2: Integrate ∫ x^2 dy = ∫ y^2 dx Step 3: Solve for y (1/3) y^3 = (1/3) x^3 + C y^3 = x^3 + C

Example 4

Solve the differential equation: dy/dx = 2xy^2 Solution: Step 1: Separate variables dy/y^2 = 2x dx Step 2: Integrate ∫ (1/y^2) dy = ∫ (2x) dx Step 3: Solve for y -1/y = x^2 + C y = -1/(x^2 + C)

Example 5

Solve the differential equation: dy/dx = x^2 / (y^2 + 1) Solution: Step 1: Separate variables (y^2 + 1) dy = x^2 dx Step 2: Integrate ∫ (y^2 + 1) dy = ∫ x^2 dx Step 3: Solve for y (1/3) y^3 + y = (1/3) x^3 + C y^3 + 3y = x^3 + 3C

These examples demonstrate how to solve differential equations using the variable separable method. It’s important to practice more problems to gain a deeper understanding of the concept.

Example 6

Solve the differential equation: dy/dx = (1 - y^2) / x Solution: Step 1: Separate variables x dy = (1 - y^2) dx Step 2: Integrate ∫ dy = ∫ (1 - y^2) / x dx Step 3: Solve for y y = √(x^2 + C) or y = -√(x^2 + C)

Example 7

Solve the differential equation: dy/dx = (x - 2y) / (x + 2y) Solution: Step 1: Separate variables (x + 2y) dy = (x - 2y) dx Step 2: Integrate ∫ (x + 2y) dy = ∫ (x - 2y) dx Step 3: Solve for y (1/2) x^2 + 2xy + C1 = (1/2) x^2 - 2xy + C2 4xy = C1 - C2 y = (C1 - C2) / 4x

Example 8

Solve the differential equation: dy/dx = e^(-2x) * y^2 Solution: Step 1: Separate variables e^(2x) dy = y^2 dx Step 2: Integrate ∫ e^(2x) dy = ∫ y^2 dx Step 3: Solve for y (1/2) e^(2x) = (1/3) y^3 + C y^3 = 3e^(2x) - 3C y = (3e^(2x) - 3C)^(1/3)

Example 9

Solve the differential equation: dy/dx = 3x^2 / (1 + y^2) Solution: Step 1: Separate variables (1 + y^2) dy = 3x^2 dx Step 2: Integrate ∫ (1 + y^2) dy = ∫ 3x^2 dx Step 3: Solve for y y + (1/3) y^3 = x^3 + C

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Example 10

Solve the differential equation: dy/dx = y(log(x))^2 Solution: Step 1: Separate variables dy / y = (log(x))^2 dx Step 2: Integrate ∫ (1/y) dy = ∫ (log(x))^2 dx Step 3: Solve for y ln|y| = (1/3) (log(x))^3 + C |y| = e^((1/3) (log(x))^3 + C) y = ±e^(C) * e^((1/3) (log(x))^3)

Summary

  • The variable separable method is used to solve ordinary differential equations.
  • The general steps are: separate variables, integrate, and solve for the variable.
  • We’ve worked through several examples using this method.
  • Practice more problems to strengthen your understanding.
  • Remember to check your solutions by substituting them back into the original differential equation.

Key Takeaways

  • Variable separable method is an important technique in integral calculus.
  • It helps to solve ordinary differential equations where variables can be separated.
  • The method involves separating variables, integrating, and solving for the variable.
  • Practice is essential to master this technique and solve complex differential equations.

Resources

Q & A

  • Any questions?
  • Let’s discuss and clarify any doubts you have.

Slide 21

  • Review of variable separable method
  • Differential equation of the form dy/dx = f(x) * g(y)
  • Steps: separate variables, integrate, solve for y

Example 11

Solve the differential equation: dy/dx = (2xy + 1) / x^2 Solution: Step 1: Separate variables x^2 dy = (2xy + 1) dx Step 2: Integrate ∫ x^2 dy = ∫ (2xy + 1) dx Step 3: Solve for y (1/3) x^3 + x = x^2y + y y = (1/3) x^3 + (1 - x^2)

Example 12

Solve the differential equation: dy/dx = y / (x + 1) Solution: Step 1: Separate variables dy / y = dx / (x + 1) Step 2: Integrate ∫ dy / y = ∫ dx / (x + 1) Step 3: Solve for y ln|y| = ln|x + 1| + C |y| = e^(ln|x + 1| + C) y = ± e^C * (x + 1)

Example 13

Solve the differential equation: dy/dx = (y - x^2) / y Solution: Step 1: Separate variables y dy = (y - x^2) dx Step 2: Integrate ∫ y dy = ∫ (y - x^2) dx Step 3: Solve for y (1/2) y^2 = (1/2) y^2 - (1/3) x^3 + C x^3 = C

Example 14

Solve the differential equation: dy/dx = y^(1/3) Solution: Step 1: Separate variables (1/y^(1/3)) dy = dx Step 2: Integrate ∫ (1/y^(1/3)) dy = ∫ dx Step 3: Solve for y 3y^(2/3) = x + C y^(2/3) = (1/3) x + C y = (1/27) x^3 + C

Example 15

Solve the differential equation: dy/dx = 1 / (x + y) Solution: Step 1: Separate variables (x + y) dy = dx Step 2: Integrate ∫ (x + y) dy = ∫ dx Step 3: Solve for y (1/2) y^2 + xy = x + C

Slide 27

  • Recap of the variable separable method
  • Practice various examples to gain proficiency
  • Differential equations can be solved using other methods too
  • Integration is a powerful tool in calculus

Summary

  • Variable separable method is useful for solving differential equations.
  • Involves separating variables, integrating, and solving for the variable.
  • Examples demonstrated the step-by-step process.
  • Use the method to solve a wide range of equations.

Key Takeaways

  • Variable separable method is applicable to differential equations.
  • Solve by separating variables, integrating, and solving for the variable.
  • Applying the method to different equations improves problem-solving skills.
  • Practice several examples to build competency.

Resources