Differential Equations - Geometry Of Homogenous Differential Equations

Differential Equations - Geometry of Homogenous Differential Equations

In this lecture, we will study the geometry of homogeneous differential equations.
We will understand the concept of homogenous differential equations and their solutions.
We will explore their graphical representation and analyze the behavior of solutions.
We will also learn about the equilibrium solutions and their stability in homogenous differential equations.
Lastly, we will discuss the phase plane analysis of homogenous differential equations.

Homogenous Differential Equations

A homogenous differential equation is of the form: $\frac{{dy}}{{dx}} = f\left(\frac{{y}}{{x}}\right)$.
Homogenous differential equations have a special property: if $y(x)$ is a solution, then $\lambda y(x)$ is also a solution for any constant $\lambda$.
We will focus on first-order homogenous differential equations in this lecture.

Solution of Homogenous Differential Equations

To solve a homogenous differential equation, we make a substitution: $y = xv$.
After substituting into the differential equation, we can solve for $v(x)$.
Once we have the solution for $v(x)$, we can substitute back to find the general solution for $y(x)$.
Let’s look at an example.

Example 1: Solve the Homogenous Differential Equation

Consider the differential equation: $\frac{{dy}}{{dx}} = \frac{{3y}}{{2x}}$.
We substitute $y = xv$ into the equation: $\frac{{dy}}{{dx}} = \frac{{xv + v}}{{x}}$.
Simplifying gives: $\frac{{xv + v}}{{x}} = \frac{{3xv}}{{2x}}$.
Cancelling out $x$, we get: $\frac{{dv}}{{dx}} + \frac{{v}}{{x}} = \frac{{3v}}{{2x}}$.
This simplifies to: $\frac{{dv}}{{dx}} = -\frac{{v}}{{2x}}$.
Now, we solve this differential equation for $v(x)$.

Solution of the Modified Differential Equation

The differential equation is now: $\frac{{dv}}{{dx}} = -\frac{{v}}{{2x}}$.
We can separate variables: $\frac{{dv}}{v} = -\frac{{1}}{{2x}}dx$.
Integrating both sides gives: $\ln|v| = -\frac{{1}}{{2}}\ln|x| + C$.
Simplifying gives: $\ln|v| = \ln|x^{-1/2}| + \ln|C|$.
Combining the logarithms, we get: $\ln|v| = \ln\left|\frac{{C}}{{\sqrt{x}}}\right|$.
Taking the exponential of both sides gives: $v = \frac{{C}}{{\sqrt{x}}}$.
This is the solution for $v(x)$.

Finding the General Solution

We substitute $v = \frac{{y}}{{x}}$ back into the solution for $v(x)$.
This gives: $\frac{{y}}{{x}} = \frac{{C}}{{\sqrt{x}}}$.
Multiplying both sides by $x$ gives: $y = C\sqrt{x}$.
This is the general solution for the homogenous differential equation: $\frac{{dy}}{{dx}} = \frac{{3y}}{{2x}}$.

Homogenous Differential Equations - Continued

Let’s look at another example to further understand the concept of homogenous differential equations.

Example 2: Solve the Homogenous Differential Equation

Consider the differential equation: $\frac{{dy}}{{dx}} = \frac{{2y}}{{x}}$.
Substitute $y = xv$ into the equation: $\frac{{dy}}{{dx}} = \frac{{xv + v}}{{x}}$.
Simplify to get: $\frac{{xv + v}}{{x}} = \frac{{2xv}}{{x}}$.
Cancel out $x$ and simplify: $\frac{{dv}}{{dx}} + \frac{{v}}{{x}} = 2v$.

Solution of the Modified Differential Equation

The differential equation is now: $\frac{{dv}}{{dx}} + \frac{{v}}{{x}} = 2v$.
We separate variables: $\frac{{dv}}{{v}} = (2 - \frac{{1}}{{x}})dx$.
Integrating both sides gives: $\ln|v| = 2x - \ln|x| + C$.
Simplify further: $\ln|v| = 2x + \ln|x^{-1}| + \ln|C|$.
Combining logarithms gives: $\ln|v| = \ln\left|Cx^2\right|$.
Taking the exponential of both sides gives: $v = Cx^2$.

Finding the General Solution

Substitute $v = \frac{{y}}{{x}}$ back into the solution for $v(x)$.
This gives: $\frac{{y}}{{x}} = Cx^2$.
Multiply both sides by $x$ gives: $y = Cx^3$.
This is the general solution for the homogenous differential equation: $\frac{{dy}}{{dx}} = \frac{{2y}}{{x}}$.

Graphical Representation of Homogenous Differential Equations

The general solution of homogenous differential equations can often be represented graphically.
The graphs can provide insights into the behavior of the solutions.
Let’s consider an example to understand the graphical representation.

Example 3: Graphical Representation

Consider the differential equation: $\frac{{dy}}{{dx}} = \frac{{6y}}{{5x}}$.
We can solve this equation for $y(x)$ to get the general solution.
The general solution is: $y = Cx^{6/5}$.

Graphical Representation - Contour Plot

We can represent the general solution of the differential equation using a contour plot.
A contour plot is a 2D representation of the solution on the x-y plane.
Each contour line represents a different value of the constant $C$.
The contour lines are curves parallel to each other.

Graphical Representation - Slope Field

Another way to represent the solution graphically is by using a slope field.
A slope field represents the slopes of the tangent lines to the solution curves at different points.
Each segment in the slope field indicates the direction and magnitude of the slope at that point.
By drawing the slope field, we can visualize the behavior of the solutions.

Graphical Representation - Solution Curves

We can also directly plot the solution curves on the x-y plane.
These curves represent the various solutions to the differential equation.
Each curve corresponds to a different value of the constant $C$.
The behavior of the curves, such as their shape and direction, can be analyzed.

Graphical Representation - Equilibrium Solutions

Equilibrium solutions are points where the derivative is zero.
For a homogenous differential equation, the only equilibrium solution is the origin (0,0).
At the origin, the slope of the tangent line is zero, indicating a point of equilibrium.
This is an important point to consider in the graphical representation.

Differential Equations - Geometry of Homogenous Differential Equations

Equilibrium Solutions

Equilibrium solutions are points where the derivative is zero.
For a homogenous differential equation, the only equilibrium solution is the origin (0,0).
At the origin, the slope of the tangent line is zero, indicating a point of equilibrium.
The equilibrium solution represents a stable point in the solution curves.
The behavior of the solution curves around the equilibrium point can be analyzed.

Stability of Equilibrium Solutions

The stability of equilibrium solutions can be determined by analyzing the behavior of the solution curves.
If the solution curves approach the equilibrium point, it is called a stable equilibrium.
If the solution curves move away from the equilibrium point, it is called an unstable equilibrium.
The stability depends on the direction of the solution curves near the equilibrium point.
This information can be obtained from the slope field or the graphical representation.

Phase Plane Analysis

Phase plane analysis is a graphical approach to analyze the behavior of solutions of a system of differential equations.
It involves plotting the solution curves on a two-dimensional plane, known as the phase plane.
The phase plane helps to visualize the trajectories and direction of motion of the solutions.
The phase plane analysis is especially useful for systems of homogenous differential equations.
It provides insights into the stability and behavior of the equilibrium solutions.

Example 4: Phase Plane Analysis

Consider the system of homogenous differential equations:
- $\frac{{dx}}{{dt}} = x - y$
- $\frac{{dy}}{{dt}} = 2x - y$
To perform a phase plane analysis, we first find the equilibrium points by setting $\frac{{dx}}{{dt}} = 0$ and $\frac{{dy}}{{dt}} = 0$.
Solving the equations gives the equilibrium point (0,0).
We can draw the phase plane by plotting the solution curves and analyzing their behavior around the equilibrium point.

Phase Plane Analysis - Solution Curves

To start, we select a set of initial conditions (x0, y0) and solve the system of differential equations.
We obtain a parametric representation of the solution curves: x(t), y(t).
By varying the initial conditions, we can plot different solution curves in the phase plane.
By analyzing the shape and behavior of these curves, we can gain insights into the dynamics of the system.

Phase Plane Analysis - Qualitative Behavior

The qualitative behavior of the solution curves provides information about the stability of the equilibrium point.
If the solution curves approach the equilibrium point, it is a stable equilibrium.
If the solution curves move away from the equilibrium point, it is an unstable equilibrium.
By analyzing the direction and movement of the solutions, we can determine the stability of the equilibrium point.
The direction of the solution curves can be obtained by analyzing the slope field or the behavior of the solutions on the phase plane.

Phase Plane Analysis - Stability

In the phase plane, stable equilibrium points are represented by attracting regions.
These regions are enclosed by solution curves that approach the equilibrium point.
Unstable equilibrium points are represented by repelling regions.
These regions are enclosed by solution curves that move away from the equilibrium point.
The stability analysis helps in understanding the long-term behavior of the system.

Phase Plane Analysis - Example Continued

Continuing with Example 4, let’s perform the phase plane analysis for the system of homogenous differential equations.
By varying the initial conditions, we can plot different solution curves in the phase plane.
Depending on the behavior of the solution curves, we can determine the stability of the equilibrium point and analyze its behavior.