Differential Equations - Positively Homogenous Domain- Definition & An Example

Differential Equations - Positively Homogeneous Domain

Definition and an Example

What is a Differential Equation?

An equation that relates a function with its derivatives.
Involves one or more variables.

What is a Homogeneous Differential Equation?

If all terms of the equation contain the dependent variable or its derivatives.

What is a Positively Homogeneous Differential Equation?

If all terms of the equation have positive integral powers of the dependent variable or its derivatives.
No negative powers or fractional powers are present.

Example of a Positively Homogeneous Differential Equation:

dy/dx = (x^2)y - x(x^2 - 1)

Steps to Solve a Positively Homogeneous Differential Equation:

Substitute y = vx into the given equation.

Differentiate both sides with respect to x.

Simplify and solve the resulting equation.

Substitute the value of v back into the equation y = vx to get the general solution.

Step 1 - Substitute y = vx into the Equation

Given differential equation: dy/dx = (x^2)y - x(x^2 - 1)
Substitute y = vx: d(vx)/dx = (x^2)(vx) - x(x^2 - 1)

Step 2 - Differentiate Both Sides with Respect to x

Differentiate left side: v + x(dv/dx)
Differentiate right side:
- Expand: x^3v - x^2 - x^3
- Differentiate: 3x^2v + x^3(dv/dx) - 2x
- Simplify: x^3(dv/dx) + 3x^2v - 2x - x^2

Step 3 - Simplify and Solve the Equation

Combine the terms with dv/dx: x(dv/dx) - x^3(dv/dx) - x^3 = -2x
Re-arrange the terms: x(dv/dx) - x^3(dv/dx) = -2x + x^3

Step 3 - Simplify and Solve the Equation (cont.)

Factor out the common term dv/dx: (x - x^3)(dv/dx) = -2x + x^3
Divide both sides by (x - x^3):
- dv/dx = (-2x + x^3)/(x - x^3)
- dv/dx = -2/(1 - x^2)

Step 4 - Substitute the Value of v back into the Equation

Recall that y = vx
Substitute v = (-2/(1 - x^2)):
- y = (-2x)/(1 - x^2)

General Solution of the Positively Homogeneous Differential Equation

The general solution is given by:
- y = (-2x)/(1 - x^2)

Properties of Positively Homogeneous Differential Equations

The sum of two homogeneous functions of the same degree is also a homogeneous function of the same degree.
The product of a homogeneous function of degree m and a homogeneous function of degree n is a homogeneous function of degree m + n.
If f(x, y) is a homogeneous function of degree n, then d/dxf(x, y) + d/dyf(x, y) is also a homogeneous function of degree n.

Example 1: Solving a Positively Homogeneous Differential Equation

Given: x(dy/dx) = y(x^2 + y^2)
Substitute y = vx:
- x(dv/dx) + v = v(x^2 + v^2)
Simplify and solve:
- x(dv/dx) = v(x^2 + v^2 - 1)
- (1/v)(dv/dx) = x/(x^2 + v^2 - 1)
- Take integral of both sides:
  - ln|v| = ln|x^2 + v^2 - 1| + ln|C|
- Exponential both sides:
  - v = C(x^2 + v^2 - 1)
- Substitute back:
  - y = vx
  - y = C(x^2 + y^2 - x^2)

Example 2: Solving a Positively Homogeneous Differential Equation

Given: y(dy/dx) - x = 0
Substitute y = vx:
- v(dv/dx)x - x = 0
Simplify and solve:
- v(dv/dx) = 1
- Take integral of both sides:
  - ln|v| = x + ln|C|
- Exponential both sides:
  - v = Ce^x
- Substitute back:
  - y = v*x
  - y = Cxe^x

Example 3: Solving a Positively Homogeneous Differential Equation

Given: x(dy/dx) = y^2 - x^2
Substitute y = vx:
- x(dv/dx) + v = v^2x^2 - x^2
Simplify and solve:
- x(dv/dx) = v^2x^2 - 2x^2
- (1/x)(dv/dx) = v^2 - 2
- Take integral of both sides:
  - ln|v^2 - 2| = x + ln|C|
- Exponential both sides:
  - v^2 - 2 = Cx
- Substitute back:
  - y = vx
  - y^2 - 2 = Cx^3

Application of Positively Homogeneous Differential Equations

Used in physics to describe physical processes.
Used in economics to model economic growth.
Used in biology to study population dynamics.
Used in engineering to solve various problems.

Solving Positively Homogeneous Differential Equations Numerically

If an exact solution cannot be obtained, numerical methods can be used.
Numerical methods approximate the solution by dividing the interval into small steps.
Popular numerical methods include Euler’s method and Runge-Kutta methods.

Euler’s Method for Numerical Approximation

Start with an initial condition (x0, y0).
Calculate the slope at the initial point (dy/dx) using the given differential equation.
Multiply the slope by a small step size h.
Add the product to the initial y-coordinate to get the next y-coordinate.
Repeat the process for the desired number of steps.

Runge-Kutta Methods for Numerical Approximation

Runge-Kutta methods involve multiple evaluations of the slope at different points.
More accurate than Euler’s method.
Popular variations include the 2nd order and 4th order Runge-Kutta methods.

Advantages of Numerical Methods

Can be used for complex differential equations that do not have exact solutions.
Can handle problems with non-linear terms and multiple variables.
Allows for evaluation and approximation of the solution at specific points.
Provides a numerical representation of the solution that can be analyzed and compared.

Summary

Positively homogeneous differential equations consist of terms that have positive integral powers of the dependent variable or its derivatives.
The steps to solve a positively homogeneous differential equation involve substitution, differentiation, simplification, and substitution of the value back into the equation.
Numerical methods can be used to approximate the solution of positively homogeneous differential equations when an exact solution is not available.

Properties of Positively Homogeneous Differential Equations (continued)

If f(x, y) is a homogeneous function of degree n, then (1/x)*f(x, y) is a homogeneous function of degree n-1.
Differentiation of a homogeneous function of degree n with respect to x gives a function of degree n-1.
Integration of a homogeneous function of degree n with respect to x gives a function of degree n+1.

Example 4: Solving a Positively Homogeneous Differential Equation

Given: x(dy/dx) = 2y
Substitute y = vx:
- x(dv/dx) + v = 2vx
Simplify and solve:
- x(dv/dx) = v(x - 2)
- (1/v)(dv/dx) = (x - 2)/x
- Take integral of both sides:
  - ln|v| = x - 2ln|x| + ln|C|
- Exponential both sides:
  - v = Cxe^(2ln|x|)
- Simplify:
  - v = Cx|x|^2
- Substitute back:
  - y = vx
  - y = Cx^2|x|

Example 5: Solving a Positively Homogeneous Differential Equation

Given: x(dy/dx) = y^2 + 2xy
Substitute y = vx:
- x(dv/dx) + v = (v^2x^2) + 2vx^2
Simplify and solve:
- x(dv/dx) = v(v^2 + 2x)
- (1/v)(dv/dx) = (v^2 + 2x)/x
- Take integral of both sides:
  - ln|v| = ln|x| + ln|v^2 + 2x| + ln|C|
- Exponential both sides:
  - v = Cx(v^2 + 2x)
- Substitute back:
  - y = vx
  - y = Cx^2(x^2 + 2x)

Example 6: Solving a Positively Homogeneous Differential Equation

Given: x(dy/dx) + y = x^2 + y^2
Substitute y = vx:
- x(dv/dx) + v = x^2 + v^2x^2
Simplify and solve:
- x(dv/dx) = (x^2 + v^2x^2) - v
- (1/x)(dv/dx) = (x + v^2x - v)/x
- Take integral of both sides:
  - ln|v| = ln|x| + ln|1 + v^2 - 1| + ln|C|
- Exponential both sides:
  - v = Cx(e^(1 + v^2 - 1))
- Simplify:
  - v = Cxe^(v^2)
- Substitute back:
  - y = vx
  - y = Cxe^(v^2)

Numerical Methods for Positively Homogeneous Differential Equations

Numerical methods provide approximate solutions to differential equations.
Euler’s method is a simple and straightforward numerical approximation method.
Runge-Kutta methods are more accurate and widely used.

Euler’s Method for Numerical Approximation (continued)

Calculate the slope at the current point using the given differential equation.
Multiply the slope by a small step size h.
Add the product to the current y-coordinate to get the next y-coordinate.
Repeat the process for the desired number of steps.

Runge-Kutta Methods for Numerical Approximation (continued)

Runge-Kutta methods involve multiple evaluations of the slope at different points.
More accurate than Euler’s method.
Popular variations include the 2nd order and 4th order Runge-Kutta methods.

Advantages of Numerical Methods (continued)

Can be used for complex differential equations that do not have exact solutions.
Can handle problems with non-linear terms and multiple variables.
Allows for evaluation and approximation of the solution at specific points.
Provides a numerical representation of the solution that can be analyzed and compared.

Summary (continued)

Positively homogeneous differential equations consist of terms that have positive integral powers of the dependent variable or its derivatives.
The steps to solve a positively homogeneous differential equation involve substitution, differentiation, simplification, and substitution of the value back into the equation.
Numerical methods can be used to approximate the solution of positively homogeneous differential equations when an exact solution is not available.

Summary (continued)

Euler’s method and Runge-Kutta methods are numerical approximation methods used to solve positively homogeneous differential equations.
Numerical methods provide an alternative approach for finding approximate solutions to differential equations.
These methods can handle complex and non-linear equations, providing valuable insights and numerical representations of the solution.