Differential Equations - Bernoulli Differential Equation & its Reduction to a Linear DE

• Bernoulli differential equation is of the form:
  • dy/dx + P(x)y = Q(x)y^n
• To solve a Bernoulli differential equation, we perform a substitution which reduces it to a linear differential equation.
• The substitution used for reduction is:
  • u = y^(1-n)
• By performing this substitution, the Bernoulli differential equation can be transformed into a linear differential equation.
• The steps for reduction of a Bernoulli differential equation to a linear differential equation are as follows:

  1. Write the given Bernoulli differential equation in standard form: dy/dx + P(x)y = Q(x)y^n

  2. Perform the substitution: u = y^(1-n)

  3. Differentiate both sides of the substitution equation with respect to x:

     • du/dx = (1-n)y^(-n)(dy/dx)

Differential Equations - Bernoulli Differential Equation & its Reduction to a Linear DE (cont.)

• Substituting this result into the original Bernoulli differential equation, we get: (1-n)y^(-n)(dy/dx) + P(x)y = Q(x)y^n

  Simplifying this equation, we get:

  (1-n)du/dx + P(x)y^(1-n) = Q(x)y^n

• Multiply the entire equation by (1-n): du/dx + (1-n)P(x)y^(1-n) = (1-n)Q(x)y^n

• The resulting equation is a linear differential equation.

• Solve this linear differential equation using the appropriate methods for linear differential equations.

• Once the solution is found, we substitute back the value of y^(1-n) to get the solution for the original Bernoulli differential equation.

• Let’s look at an example to better understand the process. Example: Solve the Bernoulli differential equation: dy/dx + 2xy = x^2y^3

Example: Bernoulli Differential Equation

• Given equation: dy/dx + 2xy = x^2y^3
• Perform the substitution: u = y^(1-3) = y^(-2)
• Differentiate both sides of the substitution equation with respect to x:
  • du/dx = -2y^(-3)(dy/dx)
• Substitute the values into the original equation:
  • (-2y^(-3)(dy/dx)) + 2xy = x^2y^3
• Multiply the equation by (-2):
  • 2y^(-3)(dy/dx) - 4xy = -2x^2y^3

Example: Bernoulli Differential Equation (cont.)

• Simplify the equation:
  • 2(dy/dx)/y^3 - 4xy = -2x^2y^3
• Rearrange the terms:
  • 2(dy/dx)/y^3 = 4xy - 2x^2y^3
• Divide both sides by 2:
  • (dy/dx)/y^3 = 2xy - x^2y^3
• This is now a linear differential equation.
• Solve this linear differential equation to find the solution for u.
• Once u is found, substitute back the value of u to get the solution for the original Bernoulli differential equation.
• Let’s solve this step by step.

Steps: Solving the Linear Differential Equation

1. Write the given linear differential equation in standard form: (dy/dx)/y^3 = 2xy - x^2y^3

2. Multiply both sides by y^3:
   • dy/dx = 2xy^4 - x^2y^6

3. Separate the variables and integrate:
   • ∫(1/y^4)dy = ∫(2xy - x^2y^3)dx

4. Evaluate the integrals on both sides:
   • (-1/3)y^(-3) = x^2y^2 - (1/3)x^3y^4 + C

5. Simplify the equation by multiplying through by 3:
   • -y^(-3) = 3x^2y^2 - x^3y^4 + 3C

Steps: Solving the Linear Differential Equation (cont.)

6. Rearrange the terms to isolate y:
   • y^(-3) - 3x^2y^2 + x^3y^4 = -3C

7. Multiply through by y^3 to get rid of the negative exponent:
   • 1 - 3x^2y^5 + x^3y^7 = -3Cy^3

8. Simplify the equation:
   • x^3y^7 - 3x^2y^5 - 3Cy^3 + y^3 - 1 = 0

9. This equation is the solution for u.

10. Substitute back the value of u = y^(-2) to get the solution for the original Bernoulli differential equation.

• The above steps demonstrate the process of solving a Bernoulli differential equation using reduction to a linear differential equation.
• Practice various examples to gain proficiency in solving Bernoulli differential equations.

Differential Equations - Bernoulli Differential Equation & its Reduction to a Linear DE

• Bernoulli differential equation is of the form: dy/dx + P(x)y = Q(x)y^n
• To solve a Bernoulli differential equation, we perform a substitution which reduces it to a linear differential equation.
• The substitution used for reduction is: u = y^(1-n)
• By performing this substitution, the Bernoulli differential equation can be transformed into a linear differential equation.
• The steps for reduction of a Bernoulli differential equation to a linear differential equation are as follows:
  1. Write the given Bernoulli differential equation in standard form: dy/dx + P(x)y = Q(x)y^n
  2. Perform the substitution: u = y^(1-n)
  3. Differentiate both sides of the substitution equation with respect to x:
     • du/dx = (1-n)y^(-n)(dy/dx)

Differential Equations - Bernoulli Differential Equation Reduction (cont.)

• Substituting this result into the original Bernoulli differential equation, we get: (1-n)y^(-n)(dy/dx) + P(x)y = Q(x)y^n

• Simplifying this equation, we get:

  (1-n)du/dx + P(x)y^(1-n) = Q(x)y^n

• Multiply the entire equation by (1-n):

  du/dx + (1-n)P(x)y^(1-n) = (1-n)Q(x)y^n

• The resulting equation is a linear differential equation.

• Solve this linear differential equation using the appropriate methods for linear differential equations.

• Once the solution is found, we substitute back the value of y^(1-n) to get the solution for the original Bernoulli differential equation.

• Let’s look at an example to better understand the process.

Example: Bernoulli Differential Equation

• Given equation: dy/dx + 2xy = x^2y^3
• Perform the substitution: u = y^(1-3) = y^(-2)
• Differentiate both sides of the substitution equation with respect to x:
  • du/dx = -2y^(-3)(dy/dx)
• Substitute the values into the original equation:
  • (-2y^(-3)(dy/dx)) + 2xy = x^2y^3
• Multiply the equation by (-2):
  • 2y^(-3)(dy/dx) - 4xy = -2x^2y^3
• Simplify the equation:
  • 2(dy/dx)/y^3 - 4xy = -2x^2y^3
• Rearrange the terms:
  • 2(dy/dx)/y^3 = 4xy - 2x^2y^3

Steps: Solving the Linear Differential Equation

1. Write the given linear differential equation in standard form: (dy/dx)/y^3 = 2xy - x^2y^3

2. Multiply both sides by y^3:
   • dy/dx = 2xy^4 - x^2y^6

3. Separate the variables and integrate:
   • ∫(1/y^4)dy = ∫(2xy - x^2y^3)dx

4. Evaluate the integrals on both sides:
   • (-1/3)y^(-3) = x^2y^2 - (1/3)x^3y^4 + C

5. Simplify the equation by multiplying through by 3:
   • -y^(-3) = 3x^2y^2 - x^3y^4 + 3C

Steps: Solving the Linear Differential Equation (cont.)

6. Rearrange the terms to isolate y:
   • y^(-3) - 3x^2y^2 + x^3y^4 = -3C

7. Multiply through by y^3 to get rid of the negative exponent:
   • 1 - 3x^2y^5 + x^3y^7 = -3Cy^3

8. Simplify the equation:
   • x^3y^7 - 3x^2y^5 - 3Cy^3 + y^3 - 1 = 0

9. This equation is the solution for u.

10. Substitute back the value of u = y^(-2) to get the solution for the original Bernoulli differential equation.

• The above steps demonstrate the process of solving a Bernoulli differential equation using reduction to a linear differential equation.
• Practice various examples to gain proficiency in solving Bernoulli differential equations.

Slide 21: Solving Bernoulli Differential Equations (Examples)

• Example 1:

  • Given equation: dy/dx - 5xy^2 = 2x^5y^8
  • Reduce the equation by substituting u = y^(1-2) = y^(-1)
  • Differentiate both sides with respect to x: du/dx = -y^(-2) * dy/dx
  • Substitute the values back into the equation and simplify.

• Example 2:

  • Given equation: dy/dx + x^2y^3 = 6xy^2
  • Reduce the equation by substituting u = y^(1-3) = y^(-2)
  • Differentiate both sides with respect to x: du/dx = -2y^(-3) * dy/dx
  • Substitute the values back into the equation and simplify.

• Example 3:

  • Given equation: dy/dx + y/x = (e^x)y^2
  • Reduce the equation by substituting u = y^(1-2) = y^(-1)
  • Differentiate both sides with respect to x: du/dx = -y^(-2) * dy/dx
  • Substitute the values back into the equation and simplify.

Slide 22: Benefits of Using Bernoulli Differential Equation

• Reduction of a Bernoulli differential equation to a linear differential equation makes it easier to solve.
• By reducing to a linear differential equation, we can utilize various methods like separation of variables, integrating factors, or variation of parameters.
• Bernoulli differential equation reduction allows us to solve a wider range of differential equations, especially those that are difficult to solve directly.
• It provides a systematic approach to solving differential equations that involve non-linear terms and power functions.
• The process of reduction helps in gaining a deeper understanding of the behavior and solutions of nonlinear differential equations.

Slide 23: Applications of Bernoulli Differential Equation

• Bernoulli differential equations have various applications in different fields of science and engineering.
• In population dynamics, the Bernoulli differential equation can model exponential growth or decay.
• In physics, Bernoulli equations are used to analyze fluid flow and aerodynamics.
• In economics, the Bernoulli differential equation can describe growth and decay in monetary systems.
• In pharmacokinetics, the Bernoulli equation can be used to model drug absorption and elimination processes.
• Bernoulli differential equations also have applications in control systems, chemistry, and biology.

Slide 24: Limitations of Bernoulli Differential Equation Reduction

• Bernoulli differential equation reduction may not be possible if the value of n is not 1 or 2.
• The reduction process can be complicated for higher values of n or when the equation includes higher order derivatives.
• There may be cases where the original Bernoulli equation cannot be reduced to a linear differential equation.
• Some Bernoulli differential equations may require additional techniques or approximations to solve, beyond the reduction process.
• It is important to carefully analyze the given equation before attempting to reduce it to a linear differential equation.

Slide 25: Summary

• Bernoulli differential equations involve nonlinear terms and can be reduced to linear differential equations using a substitution method.
• The substitution used is u = y^(1-n).
• By substituting this value into the original equation and simplifying, the Bernoulli equation becomes a linear differential equation.
• Solving the resulting linear differential equation provides the solution for u.
• Substituting the value of u back into the original substitution equation gives the solution for the original Bernoulli differential equation.
• Bernoulli differential equations have various applications and can be solved using reduction techniques for easier analysis.

Slide 26: Additional Resources

• Differential Equations: An Introduction by H. T. Davis
• Differential Equations with Applications and Historical Notes by George F. Simmons
• Differential Equations for Dummies by Steven Holzner
• Khan Academy: Differential Equations
• MIT OpenCourseWare: Ordinary Differential Equations

Slide 27: Q&A

• Are there any questions about the reduction of Bernoulli differential equations?
• Can you provide an example of a Bernoulli differential equation that cannot be reduced to a linear differential equation?
• What are some real-world applications of Bernoulli differential equations?
• Are there any alternative methods to solve Bernoulli differential equations?
• How can we determine if a given differential equation is a Bernoulli equation?

Slide 28: References

• Boyce, W. E., & DiPrima, R. C. (2012). Elementary differential equations and boundary value problems. Wiley.
• Nagle, R. K., Saff, E. B., & Snider, A. D. (2016). Fundamentals of differential equations. Pearson Education.
• Tenenbaum, M., & Pollard, H. (2018). Ordinary differential equations. Courier Dover Publications.

Slide 29: Thank You

• Thank you for attending this lecture on Bernoulli differential equations and their reduction to linear differential equations.
• If you have any further questions or need clarification, please feel free to reach out.
• Stay tuned for more lectures on differential equations and other topics in mathematics.
• Good luck with your studies and upcoming exams!