Notes/Calculus 2/Linear ODEs.md

date	type
09.09.2024	math

Method of Variation of Constants

The method of variation of constants is a technique used to find a particular solution to a non-homogeneous linear differential equation. This method generalizes the solution of homogeneous equations by allowing the constants in the general solution to vary as functions of the independent variable.

Notice how it's similar to Recurrence relations

1. Solve the homogeneous equation: Start by solving the associated homogeneous differential equation. For an ODE of the form:

   
   y'' + p(x)y' + q(x)y = g(x),
   

   solve the homogeneous part:

   
   y'' + p(x)y' + q(x)y = 0.
   

   The general solution to the homogeneous equation will be:

   
   y_h(x) = C_1 y_1(x) + C_2 y_2(x),
   

   where y_1(x) and y_2(x) are linearly independent solutions.

2. Replace constants with functions: Replace the constants C_1 and C_2 with functions u_1(x) and u_2(x):

   
   y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x).
   

3. Set up equations for u_1(x) and u_2(x): Differentiate y_p(x) and use the condition that u_1'(x)y_1(x) + u_2'(x)y_2(x) = 0 to avoid second derivatives of u_1(x) and u_2(x). This gives:

   
   u_1'(x)y_1(x) + u_2'(x)y_2(x) = 0,
   

   
   u_1'(x)y_1'(x) + u_2'(x)y_2'(x) = g(x).
   

4. Solve for u_1'(x) and u_2'(x): Solve this system of equations to find u_1'(x) and u_2'(x).

5. Integrate to find u_1(x) and u_2(x): Integrate to find u_1(x) and u_2(x).

6. Form the particular solution: The particular solution is:

   
   y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x).
   

Bernoulli Equation

A Bernoulli equation is a type of first-order nonlinear differential equation of the form:

\frac{dy}{dx} + P(x)y = Q(x)y^n,

where n \neq 0, 1.

• When and how do we apply it?

To solve a Bernoulli equation:

1. Divide through by y^n:

   
   y^{-n} \frac{dy}{dx} + P(x)y^{1-n} = Q(x).
   

2. Make a substitution: Let v = y^{1-n}. Then \frac{dv}{dx} = (1-n)y^{-n} \frac{dy}{dx}.

3. Rewrite the equation in terms of v:

   
   \frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x).
   

   This is now a linear differential equation in v(x).

4. Solve the linear ODE for v:

   Use an integrating factor to solve for v(x).

5. Substitute back to find y(x):

   Since v = y^{1-n}, solve for y(x).

Riccati Equation

A Riccati equation is a first-order nonlinear differential equation of the form:

\frac{dy}{dx} = a(x) + b(x)y + c(x)y^2.

• When do we use it?

Riccati equations are used in various fields such as control theory and fluid dynamics. They can sometimes be solved by making an appropriate substitution if a particular solution is known. In general, Riccati equations do not have a straightforward general solution like linear ODEs.

n \geq 2 Linear ODE

• What do we do with those?

For linear ODEs of order n \geq 2, we typically look for a general solution that is a linear combination of n linearly independent solutions.

General Properties of Spaces of Solutions of such \epsilon

• Linear dependence: Solutions y_1(x), y_2(x), \ldots, y_n(x) are linearly independent if no solution can be written as a linear combination of the others.

• Dimension: The solution space of a linear homogeneous ODE of order n has dimension n.

• Fundamental theorem: If y_1(x), y_2(x), \ldots, y_n(x) are n linearly independent solutions to an $n$-th order linear homogeneous ODE, then any solution can be written as:

  
  y(x) = C_1 y_1(x) + C_2 y_2(x) + \cdots + C_n y_n(x),
  

  where C_1, C_2, \ldots, C_n are constants.

• Structure of the space: The space of solutions is a vector space, where each solution can be represented as a linear combination of a set of basis solutions.