Theory of Linear Equations

II.1 Theory of Linear Equations

1. Introduction to Linear Differential Equations

• This lecture extends the understanding of differential equations, specifically focusing on linear differential equations with initial conditions.

2. Initial Value Problem (IVP)

• Definition:

  • An nth order linear initial value problem can be expressed as: $a<em>n(x) \frac{d^n y}{dx^n} + a</em>{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + … + a<em>1(x) \frac{dy}{dx} + a</em>0(x)y = g(x)$

    • With initial conditions:

    • $y(x0) = y0$

    • $y'(x0) = y1$

    • …

    • $y^{(n-1)}(x0) = y{n-1}$

  • Where I is an open interval such that $x_0 \in I$.

  • Functions $an(x), a{n-1}(x), …, a_0(x), g(x)$ are continuous on I.

2.1 Existence and Uniqueness of Solutions

• The problem has a unique solution on interval I that satisfies the corresponding initial conditions.

  • Uniqueness is guaranteed under the following conditions:

    1. The functions involved are continuous.

    2. The equality holds.

    3. The initial conditions are specified at one point.

2.2 Example 1

• Example Problem:

  • $3y'' + 5y' - y + 7y = 0$

  • Initial Conditions:

    • $y(1) = 0$, $y'(1) = 0$, $y''(1) = 0$

  • This represents a third-order linear initial value problem. Unique solution: $y = 0$ (the only solution).

2.3 Example 2

• Example Problem:

  • $4y'' + (4 - 44)y' = 12x$

  • Initial Conditions:

    • $y(0) = 0$, $y'(0) = 1$

  • Here this is a second-order linear IVP with all functions continuous on some interval containing $x_0 = 0$. There is a unique solution.

3. Boundary Value Problems (BVP)

• Definition:

  • A common type of problem where the values of $y$ and its derivatives are specified at different points, rather than at a single point.

  • General formulation: $a<em>n(x) y^{(n)} + a</em>{n-1}(x) y^{(n-1)} + … + a_0(x)y = g(x)$

    • With conditions defined at boundary points $x = a$ and $x = b$.

3.1 Example of BVP

• Example Problem:

  • General formulation of a classic BVP:
    $y(a) = y<em>0, y(b) = y</em>1$

  • Conditions specified at points other than just one:

    • This means we have specified not just $y(a)$ but also $y'(b)$ and potentially other values.

• BVP can result in:

  • No solution

  • One solution

  • Infinitely many solutions depending on the specific equations and conditions.

3.2 Example of Periodic Boundary Conditions

• Example Boundary Problem: Consider $y'' + 16y = 0$

  • If we take conditions such as:

    1. $y(0) = 0$

    2. $y( rac{
       ho}{2}) = 0$

  • Leads to infinitely many solutions due to conditions in the periodic boundary setup.

4. Homogeneous Differential Equations and Superposition

4.1 Definition

• A linear differential equation of the form: $a<em>n(t) y^{(n)} + … + a</em>0(t)y = g(t)$

  • Called homogeneous if $g(t) = 0$, and non-homogeneous if $g(t) \neq 0$.

4.2 Previous Unit Recap

• Previously studied linear, inhomogeneous problems can be represented generally as:

  • $y' + P(t)y = g(t)$

  • General solution can be written as: $y(t) = y<em>h(t) + y</em>p(t)$

    • Where $y_h(t)$ is the complementary solution to the associated homogeneous equation.

4.3 Associated Homogeneous Problem

• Dividing the expression by $a_n(t)$ gives the associated homogeneous problem:

  • $[a_n(t)y'' + … = 0]$

  • Results in a solution set.

4.4 Property of Homogeneous Codes

• If $y1(t)$ and $y2(t)$ are solutions to the homogeneous equation, so is:

  • $C<em>1 y</em>1(t) + C<em>2 y</em>2(t)$

  • This applies to any linear combination of solutions, showcasing linearity and superposition.

5. Linearly Independent Functions

• Definition:

  • Functions $f1(t), f2(t), …, fk(t)$ are linearly independent if the only solution to the equation: $C</em>1 f<em>1(t) + C</em>2 f<em>2(t) + … + C</em>k f_k(t) = 0$

    • Is $C1 = C2 = … = C_k = 0$.

  • Conversely, if nontrivial linear combinations exist, the functions are linearly dependent.

5.1 Example of Linear Independence

• Example:

  • Consider functions:

    • $f_1(t) = 3 ext{cos}(t) + 5x^2$

    • $f_2(t) = 5x^2 + 3x$

  • These are dependent as they result in nontrivial solutions.

6. Wronskian Determinant

• To check linear independence, use the Wronskian determinant defined as follows:

  • If $f1, f2, ext{…}, f_k$ are functions, prepare the Wronskian:

  • $$ W(f1, f2, …, fk)(x) = egin{vmatrix} f1 & f2 & … & fk \ f'1 & f'2 & … & f'k \ … & f^{(k-1)}1 & f^{(k-1)}2 & … & f^{(k-1)}k \ ext{…} \ ext{…}egin{vmatrix} ext{determinant here} ext{determinant section} \ ext{…}\