Differential Equations and Numerical Methods Review

Solutions and Order of Differential Equations

A differential equation is an equation that involves a function $y$ and one or more of its derivatives. To determine if a given function is a solution to a specific differential equation, one must perform a verification process involving differentiation and substitution.

Verification of Solutions

To check if a possible solution satisfies a differential equation, follow these steps:

1. Identify the derivatives present in the differential equation (e.g., $y', y'', y'''$).
2. Calculate the corresponding derivatives of the candidate function.
3. Substitute the function and its derivatives into the original differential equation.
4. Simplify the algebraic expression to determine if the equality holds true.

Detailed Example

Consider the differential equation: $y'' - 3y' + 2y = 24e^{-2x}$

Candidate solution: $y = 3e^x + 4e^{2x} + 2e^{-2x}$

Step 1: Compute Derivatives

• First Derivative ($y'$):   $y' = 3e^x + 8e^{2x} - 4e^{-2x}$
• Second Derivative ($y''$):   $y'' = 3e^x + 16e^{2x} + 8e^{-2x}$

Step 2: Substitution into the Left-Hand Side (LHS) Substitute $y, y', y''$ into $y'' - 3y' + 2y$: $(3e^x + 16e^{2x} + 8e^{-2x}) - 3(3e^x + 8e^{2x} - 4e^{-2x}) + 2(3e^x + 4e^{2x} + 2e^{-2x})$

Step 3: Distribution and Grouping

• $e^x$ terms: $3 - 9 + 6 = 0$
• $e^{2x}$ terms: $16 - 24 + 8 = 0$
• $e^{-2x}$ terms: $8 + 12 + 4 = 24$

Result: $24e^{-2x}$. Since the LHS equals the Right-Hand Side (RHS), the candidate is a solution.

The Order of a Differential Equation

The order of a differential equation is defined by the highest order of derivative that appears in the equation. This concept is analogous to the degree of a polynomial.

• First Order: Contains at most a first derivative ($y'$).
• Third Order: Contains at most a third derivative ($y'''$).
• Higher Order Notation: After the third derivative, the use of primes (apostrophes) becomes impractical. Instead, numerical notation is used inside parentheses, such as $y^{(4)}$ for the fourth derivative or $y^{(10)}$ for the tenth derivative. This notation distinguishes the derivative order from an exponent (e.g., $y^4$ vs. $y^{(4)}$).

General and Particular Solutions

General Solutions

A general solution describes all possible functions that satisfy a differential equation. It typically includes one or more constants of integration (e.g., $+ C$). Because constants disappear during differentiation, they do not affect the validity of the solution relative to the derivative relationship.

• Example: Given $y' = 2x$, the general solution is $y = x^2 + C$.

Particular Solutions and Initial Value Problems (IVP)

A particular solution is a specific function from the general family that satisfies a given set of initial conditions (usually a point the function must pass through).

• Initial Condition: A constraint like $y(x_0) = y_0$.
• Procedure: Find the anti-derivative (integrate), then substitute the initial condition to solve for the unknown constant $C$.

Example: Solving an IVP

Solve $y' = 2 - x$ passing through the point $(2, 7)$.

1. Find the anti-derivative: $y = 2x - \frac{x^2}{2} + C$
2. Substitute the point: $7 = 2(2) - \frac{(2)^2}{2} + C$
3. Solve for $C$: $7 = 4 - 2 + C \rightarrow C = 5$
4. Particular solution: $y = 2x - \frac{x^2}{2} + 5$

Kinematic Modeling with Differential Equations

Differential equations are fundamentally used to model real-world scenarios where the rate of change of a variable is known. A classic example is projectile motion under the influence of gravity.

Problem Data

• Scenario: A baseball is thrown upward from a height of $3\,m$ with an initial velocity of $10\,m/s$.
• Force: The only force acting is gravity, providing a constant acceleration of approximately $-9.8\,m/s^2$.
• Constants: Mass = $0.15\,kg$ (note: mass is irrelevant in this vacuum-gravity model).

Determining Velocity ($v(t)$)

1. Acceleration is the derivative of velocity: $v'(t) = -9.8$
2. Integrate to find velocity: $v(t) = -9.8t + C$
3. Use initial velocity $v(0) = 10$: $10 = -9.8(0) + C \rightarrow C = 10$
4. Velocity function: $v(t) = -9.8t + 10$
   • To find velocity at $t = 2\,s$: $v(2) = -9.8(2) + 10 = -9.6\,m/s$ (the ball is moving downward).

Determining Position ($s(t)$)

1. Velocity is the derivative of position: $s'(t) = -9.8t + 10$
2. Integrate to find position: $s(t) = -4.9t^2 + 10t + C$
3. Use initial height $s(0) = 3$: $3 = -4.9(0) + 10(0) + C \rightarrow C = 3$
4. Position function: $s(t) = -4.9t^2 + 10t + 3$

Qualitative Analysis: Direction Fields

When a differential equation cannot be solved easily using anti-derivatives (e.g., when it involves multiple variables or complex products), we use Direction Fields (or Slope Fields) to visualize the behavior of the solutions.

Concept of the Direction Field

A differential equation $y' = f(x, y)$ provides the slope of the tangent line at any point $(x, y)$. By drawing small arrows with these slopes across a grid, we can trace the path a solution curve would take.

Equilibrium Solutions

Equilibrium solutions are horizontal lines where the rate of change is zero ($y' = 0$). They represent states where the system is "stable."

• Example: $y' = 0.4y(y - 72)$
  • If $y = 72$, then $y' = 0$. This is an equilibrium solution.
  • Behavior: If the initial condition is above 72, the negative slope pushes the solution down toward 72. If below 72 but positive, the positive slope pushes it up toward 72.

Complex Slope Fields

For equations like $y' = 3x + 2y - 4$, the slope depends on both $x$ and $y$.

• To analyze: Plug in arbitrary points (e.g., $(0,0), (1,0), (0,1)$) and calculate the value of $y'$ to determine the tilt of the arrows.
• Patterns emerge as you move across the axes; for instance, as $x$ increases, the slope may become more positive, creating parabolic-like behavior or specific "flow" patterns similar to fluid dynamics.

Numerical Methods: Euler's Method

Euler's method is a numerical procedure used to approximate the solution curve of a differential equation when exact integration is not possible. It relies on the principle of Linearization (linear approximation).

The Linearization Formula

The linearization of $f$ at a point $a$ is: $L(x) = f'(a)(x - a) + f(a)$

The Methodology

1. Start at a known point $(x_0, y_0)$.
2. Use the differential equation to calculate the slope at that point: $m = y'(x_0, y_0)$.
3. Move a small step, $h$ (the "step size"), along the tangent line to reach a new point $(x_1, y_1)$.
4. Repeat the process from the new point.

Iterative Step Breakdown

Example: $y' = x^2 - y^2$ starting at $(-1, 2)$.

• Step 1: Calculate slope at $(-1, 2)$: $y' = (-1)^2 - (2)^2 = -3$.
• Step 2: Formulate linearization: $L(x) = -3(x - (-1)) + 2 = -3(x + 1) + 2$.
• Step 3: Predict new value for $x = -0.9$:   $y_{new} = -3(-0.9 + 1) + 2 = 1.7$
• Step 4: Calculate the new slope at $(-0.9, 1.7)$: $y' = (-0.9)^2 - (1.7)^2 = -2.08$.
• Step 5: Formulate the new linearization and repeat.

Note on Accuracy: Euler's method involves compounding errors (estimations based on estimations). The accuracy is highly dependent on the step size; smaller steps lead to more accurate models. In modern practice, this is typically handled by computers (e.g., Python programs) rather than manual calculation.

Questions & Discussion

• Relationship between Order and Degree: A student compared the order of a derivative to the degree of a polynomial. The speaker confirmed that they are similar conceptually—representing the "highest level" of the operation present.
• Non-Integer Derivatives: A student asked if it is possible to take a "half derivative." The instructor noted that in standard differential equations, we only deal with discrete, positive integer orders (1, 2, 3…), though context usually prevents confusion between powers and orders.
• Application in Physics: The discussion touched upon how the mass of the baseball was provided but did not affect the kinematic equations of motion. It was also noted that while velocity is the derivative of position and acceleration is the derivative of velocity, the derivative of acceleration (known as jerk) is rarely discussed at this introductory level and may appear in Physics 2 or 3.
• Programming Euler's Method: It was acknowledged that performing hundreds of iterations of Euler's method by hand is "miserable," suggesting that these methods are perfect candidates for simple computational scripts.