Differential Equations - Homogeneous Differential Equation

• In this lecture, we will study homogeneous differential equations and their properties.
• A homogeneous differential equation is of the form dy/dx = f(x,y) where f(x,y) is a homogeneous function.
• A homogeneous function is a function for which f(kx, ky) = k^n * f(x, y), for some constant k.

Properties of Homogeneous Differential Equations

1. If the degree of f(x,y) is n, then the degree of f(x,y)/x is n-1.

2. If f(x,y) is a homogeneous function of degree n, then g(x,y)= f(x,y)/y is a homogeneous function of degree n-1.

3. If y = u/v is a solution of the homogeneous differential equation, then y = ku/kv is also a solution, where k is a constant.

4. The equation dx/P = dy/Q, where P and Q are homogeneous functions of the same degree, is a homogeneous differential equation.

Example 1:

• Solve the homogeneous differential equation: dy/dx = (2x^3+3xy^2)/(3x^2y+4y^3) Solution:

  • Divide both sides of the equation by x^3: (1/x^3)dy/dx = (2+3(y/x)^2)/(3(y/x)+4(y/x)^3)

  • Let u = y/x, then dy/dx = u’ + u/x

  • Substituting the values back in the equation: (1/x^3)(u’ + u/x) = (2+3u^2)/(3u+4u^3)

  • Rearranging the equation: x^2u’ + xu = (2+3u^2)/(3u+4u^3)

  • Simplifying further, we get: u’ + (1/x)u = (2+3u^2)/(3u+4u^3)

Example 2:

• Solve the homogeneous differential equation: (x^2+y^2)dx - 2xydy = 0 Solution:

  • Let u = y/x, then dy/dx = u’ + u/x

  • Substitute the values in the equation: (x^2+(ux)^2)dx - 2x(ux)dy = 0

  • Simplify further: x^2(1+u^2)dx - 2x^2udy = 0

  • Divide both sides by x^2 and rearrange: (1+u^2)dx - 2udy = 0

  • Rearranging the terms further, we get: (1+u^2)dx = 2udy

Differential Equations - Homogeneous Differential Equations

• In this lecture, we will dive deeper into homogeneous differential equations and discuss their properties.
• A homogeneous differential equation is of the form dy/dx = f(x,y) where f(x,y) is a homogeneous function.
• A homogeneous function is a function for which f(kx, ky) = k^n * f(x, y), for some constant k.
• Homogeneous differential equations have several interesting properties that we will explore.

Properties of Homogeneous Differential Equations

1. If the degree of f(x,y) is n, then the degree of f(x,y)/x is n-1.

2. If f(x,y) is a homogeneous function of degree n, then g(x,y)= f(x,y)/y is a homogeneous function of degree n-1.

3. If y = u/v is a solution of the homogeneous differential equation, then y = ku/kv is also a solution, where k is a constant.

4. The equation dx/P = dy/Q, where P and Q are homogeneous functions of the same degree, is a homogeneous differential equation.

Example 1: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: dy/dx = (2x^3+3xy^2)/(3x^2y+4y^3) Solution:

  • Divide both sides of the equation by x^3: (1/x^3)dy/dx = (2+3(y/x)^2)/(3(y/x)+4(y/x)^3)
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substituting the values back in the equation: (1/x^3)(u’ + u/x) = (2+3u^2)/(3u+4u^3)
  • Rearranging the equation: x^2u’ + xu = (2+3u^2)/(3u+4u^3)
  • Simplifying further, we get: u’ + (1/x)u = (2+3u^2)/(3u+4u^3)

Example 2: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (x^2+y^2)dx - 2xydy = 0 Solution:

  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (x^2+(ux)^2)dx - 2x(ux)dy = 0
  • Simplify further: x^2(1+u^2)dx - 2x^2udy = 0
  • Divide both sides by x^2 and rearrange: (1+u^2)dx - 2udy = 0
  • Rearranging the terms further, we get: (1+u^2)dx = 2udy

Homogeneous Differential Equations - Properties Continued

5. Homogeneous differential equations can be made into separable equations by making the substitution y = vx.

6. The general solution of a homogenous linear differential equation is given by y = x^m * (C1 + C2 ln|x|), where m is the degree of homogeneity and C1, C2 are constants.

7. If f(x, y)dx + g(x, y)dy = 0 is a homogeneous differential equation, then the change of variables v = y/x converts it to an exact differential equation dv = (g(x, y) - f(x, y)/x)dx.

Homogeneous Differential Equations - Properties Continued

8. If y = v * x is a solution of the homogeneous differential equation, where v is a function of x, then dy/dx = v + x * dv/dx.

9. Homogeneous differential equations can sometimes be solved using the substitution y = ux or x = vy.

10. The solution to a homogeneous differential equation can be found by writing it as a separable equation, integrating and solving for y.

Example 3: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (2x^3y + y^4) dx - (3x^4 + 4xy^3) dy = 0 Solution:

  • Divide both sides by y^4 and rearrange: (2x^3/y^3 + 1) dx - (3x^4/y^4 + 4x/y) dy = 0
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (2u^3+1)dx - (3u^4+4u)dy = 0
  • Rearranging the terms further, we get: (2u^3+1)/u^4 dx - (3u^3+4)dy = 0
  • This equation is now separable, integrate and solve for u.

Example 4: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (x^2+y^2) dy/dx = (3xy-y^2)/(x^2-y^2) Solution:

  • Multiply both sides by (x^2 - y^2): (x^2+y^2) dx = (3xy-y^2)dy/dx * (x^2-y^2)
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (x^2+(ux)^2) dx = (3x(ux)-(ux)^2)dy/dx * (x^2-(ux)^2)
  • Simplify further: (1+u^2) dx = 3u*x - u^2 dx/dy * (1-u^2)
  • Rearrange the terms: (1+u^2)/(3u*x - u^2) dx = dx/dy

Example 5: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (x^2+y^2) dx - 2xy dy = 0 Solution:

  • Divide both sides by x^2: (1+(y/x)^2)dx - 2(y/x)dy = 0
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (1+u^2)dx - 2udy = 0
  • Rearranging further, we get: (1+u^2)dx = 2udy
  • This equation can be solved directly by integrating both sides.

Summary

• Homogeneous differential equations have the form dy/dx = f(x,y), where f(x,y) is a homogeneous function.
• Homogeneous functions have specific properties that make them useful in solving differential equations.
• We discussed properties of homogeneous differential equations and demonstrated their application through examples.
• Further exploration into solving specific types of homogeneous differential equations will be covered in future lectures.

Differential Equations - Homogeneous Differential Equations

Summary (Continued)

• Homogeneous differential equations have special properties that can be leveraged to solve them.
• Making appropriate substitutions and rearranging terms can transform a homogeneous differential equation into a more solvable form.
• The method of separation of variables can often be used to solve homogeneous differential equations.
• Specific techniques, such as substitutions and change of variables, can also be employed to solve certain types of homogeneous differential equations.
• The general solution to a homogeneous linear differential equation can be expressed in terms of a power function and logarithm.
• Homogeneous differential equations can sometimes be converted into exact differential equations, facilitating their solution.

Example 6: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (3xy^2-2x^2y) dx + (x^2y-3x^3) dy = 0 Solution:

  • Divide through by x^3: (3y^2-2xy) dx + (xy/x^2-3x) dy = 0
  • Divide through by x^3 and y: (3(y/x)^2-2(y/x)) dx + ((y/x)-(3x/y)) dy = 0
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (3u^2-2u) dx + (u-3x/u) dy = 0
  • Rearrange the terms further: (3u^2-2u) dx = (3x/u-u) dy
  • This equation is now separable, integrate and solve for u.

Example 7: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (x^2+y^2) dy/dx + 2xy = 0 Solution:

  • Divide through by x^2: (1+(y/x)^2) dy/dx + 2(y/x) = 0
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (1+u^2) (u’ + u/x) + 2u = 0
  • Simplify further: u’ + (1/x)u + u^3 + 2u = 0
  • Rearrange the terms: u’ + (1/x)u + u(1+u^2) + 2u = 0
  • This equation is not separable, but it can be solved using other techniques.

Further Techniques for Solving Homogeneous Differential Equations

• Substitution y = ux or x = vy can sometimes simplify the equation and facilitate its solution.
• If the given homogeneous differential equation is in the form dx/P = dy/Q, where P and Q are both homogeneous functions of the same degree, the change of variables v = y/x can convert it into an exact differential equation.

Further Techniques for Solving Homogeneous Differential Equations

• If the given equation is not directly solvable, we can try multiplying by an integrating factor to make it exact.
• For some types of homogeneous differential equations, using polar coordinates can lead to a simpler form and an easier solution.
• The solution of a homogeneous differential equation can sometimes be found by writing it as a separable equation, integrating, and solving for y.

Example 8: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (2xy+x^2y^2) dy/dx = y^3(1-xy) Solution:

  • Divide through by y^3: (2x/y+x^2/y^2) dy/dx = 1-xy
  • Divide through by y^3 and x: (2(y/x)+x/y) dy/dx = 1-xy
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (2u+xu’) + (1-ux) = 0
  • This equation is not separable, but it can be simplified further by rearranging terms.

Example 9: Solving a Homogeneous Differential Equation

• Solve the homogeneous differential equation: (x^2+y^2) dy/dx = y(3x-y^2) Solution:

  • Divide through by y(x^2+y^2): (1/y) dy/dx = (3x-y^2)/(x^2+y^2)
  • Let u = y/x, then dy/dx = u’ + u/x
  • Substitute the values in the equation: (1/u) (u’ + u/x) = (3-u^2)/(1+u^2)
  • This equation can be solved by separation of variables and further integration.

Conclusion

• Homogeneous differential equations are a specific type of differential equation in which the equation can be represented in the form dy/dx = f(x,y), where f(x,y) is a homogeneous function.
• Homogeneous differential equations possess certain properties that can be exploited to solve them.
• Making appropriate substitutions, rearranging terms, and using techniques like separation of variables, change of variables, and integration can aid in solving these equations.
• Examples have been provided to illustrate the process of solving homogeneous differential equations.
• Further exploration into the topic will include more specialized techniques for solving specific types of homogeneous differential equations.

Summary

• Homogeneous differential equations possess unique characteristics that set them apart from other types of differential equations.
• Understanding the properties and techniques for solving homogeneous differential equations is important for successful problem-solving.
• By leveraging properties such as homogeneity and making suitable substitutions, the process of solving these equations can be simplified.
• Examples have been presented to reinforce concepts and demonstrate problem-solving strategies.
• Further practice and exploration of various types of homogeneous differential equations will enhance understanding and mastery of this topic.

Thank You!

• Please feel free to ask any questions.