27-01 — Introduction to Differential Equations

Phase: Ordinary Differential Equations | Subject: 27-01 Prerequisites: 05-01-antiderivatives.md, 04-04-differentiation-rules.md Next subject: 27-02-first-order-odes.md

Learning Objectives

By the end of this subject, you will be able to:

1. Define ordinary and partial differential equations
2. Classify DEs by order, linearity, homogeneity, and autonomous vs nonautonomous
3. Interpret DEs geometrically via slope fields
4. Distinguish general, particular, and singular solutions
5. Verify that a given function satisfies a differential equation

Core Content

What Is a Differential Equation?

A differential equation (DE) is an equation involving an unknown function and its derivatives.

Ordinary Differential Equation (ODE): unknown function depends on ONE independent variable.

$$\frac{dy}{dx} = ky \quad \text{or} \quad y'' + \omega^2 y = 0$$

Partial Differential Equation (PDE): unknown function depends on MULTIPLE variables.

$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \quad \text{(heat equation)}$$

Classification

1. Order: the highest derivative appearing. - $\frac{dy}{dx} = 3y$ → 1st order - $y'' + 2y' + y = 0$ → 2nd order - $y''' = \sin x$ → 3rd order

2. Linearity: - Linear: $a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \cdots + a_0(x) y = g(x)$ - The dependent variable $y$ and its derivatives appear to the first power only - No products like $y y'$, no functions of $y$ like $\sin y$ - Nonlinear: anything else. Examples: $y' = y^2$, $y'' + \sin y = 0$ (pendulum)

3. Homogeneous vs Nonhomogeneous: - Homogeneous: $g(x) = 0$ in the linear form above - Nonhomogeneous: $g(x) \neq 0$

4. Autonomous vs Nonautonomous: - Autonomous: independent variable does not appear explicitly: $y' = f(y)$ - Nonautonomous: $y' = f(x, y)$

Solutions

• General solution: contains $n$ arbitrary constants for an $n$-th order ODE
• Particular solution: obtained by specifying initial/boundary conditions
• Singular solution: not obtainable from the general solution by choosing constants

Example: $y' = y$. General solution: $y = Ce^x$. Particular: $y = 2e^x$ (with $y(0) = 2$).

Initial Value Problem (IVP)

An $n$-th order ODE with $n$ initial conditions:

$$y^{(n)} = f(x, y, y', \ldots, y^{(n-1)}), \quad y(x_0) = y_0, y'(x_0) = y_1, \ldots, y^{(n-1)}(x_0) = y_{n-1}$$

Existence and Uniqueness: Under mild conditions (continuity of $f$ and Lipschitz condition), an IVP has exactly one solution.

Direction Fields (Slope Fields)

For $y' = f(x, y)$, draw a short line segment with slope $f(x, y)$ at each point $(x, y)$. Solution curves follow these slopes.

This provides geometric intuition for qualitative behavior even when analytical solutions are hard.

Key Terms

• Ordinary Differential Equation (ODE)
• Partial Differential Equation (PDE)
• Order, Linear, Nonlinear
• Homogeneous, Nonhomogeneous
• General Solution, Particular Solution
• Initial Value Problem (IVP)

Worked Examples

Example 1: Classification

Classify $x^2 y'' + xy' + (x^2 - 1)y = 0$ (Bessel's equation).

Solution: - Order: 2 (highest derivative is $y''$) - Linear: Yes (no products of $y$ terms, $y$ and derivatives appear to power 1) - Homogeneous: Yes (right-hand side is 0) - Coefficients: Variable ($x^2, x, x^2-1$ depend on $x$)

Example 2: Verify a Solution

Verify that $y = e^{-x} + 2e^{3x}$ satisfies $y'' - 2y' - 3y = 0$.

Solution: $y = e^{-x} + 2e^{3x}$ $y' = -e^{-x} + 6e^{3x}$ $y'' = e^{-x} + 18e^{3x}$

Plug in: $y'' - 2y' - 3y = (e^{-x} + 18e^{3x}) - 2(-e^{-x} + 6e^{3x}) - 3(e^{-x} + 2e^{3x})$ $= e^{-x} + 18e^{3x} + 2e^{-x} - 12e^{3x} - 3e^{-x} - 6e^{3x} = 0$. ✓

Example 3: Slope Field Interpretation

For $y' = x - y$, sketch the isocline where $y' = 0$ and determine the qualitative behavior.

Solution: $y' = 0$ when $y = x$. Above this line, $y' < 0$ (decreasing). Below this line, $y' > 0$ (increasing). Solutions approach the line $y = x - 1$ as $x \to \infty$. One particular solution is exactly $y = x - 1$.

Quiz

Q1: What does the concept of Ordinary Differential Equation (ODE) primarily refer to in this subject?

A) A historical anecdote about Ordinary Differential Equation (ODE) B) A computational error related to Ordinary Differential Equation (ODE) C) A visual representation of Ordinary Differential Equation (ODE) D) The definition and application of Ordinary Differential Equation (ODE)

Correct: D)

• If you chose A: This is incorrect. Ordinary Differential Equation (ODE) is defined as: the definition and application of ordinary differential equation (ode). The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Ordinary Differential Equation (ODE) is defined as: the definition and application of ordinary differential equation (ode). The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Ordinary Differential Equation (ODE) is defined as: the definition and application of ordinary differential equation (ode). The other options describe different aspects that are not the primary focus.
• If you chose D: Ordinary Differential Equation (ODE) is defined as: the definition and application of ordinary differential equation (ode). The other options describe different aspects that are not the primary focus. Correct!

Q2: Which of the following is the key formula discussed in this subject?

A) A simplified version of \frac{dy}{dx} = ky \quad \t... B) \frac{dy}{dx} = ky \quad \text{or} \quad y'' + \omega^2 y = 0 C) The inverse operation of the formula in question D) An unrelated formula from a different topic

Correct: B)

• If you chose A: This is incorrect. The formula \frac{dy}{dx} = ky \quad \text{or} \quad y'' + \omega^2 y = 0 is central to this subject. The other options are either simplified versions or unrelated.
• If you chose B: The formula \frac{dy}{dx} = ky \quad \text{or} \quad y'' + \omega^2 y = 0 is central to this subject. The other options are either simplified versions or unrelated. Correct!
• If you chose C: This is incorrect. The formula \frac{dy}{dx} = ky \quad \text{or} \quad y'' + \omega^2 y = 0 is central to this subject. The other options are either simplified versions or unrelated.
• If you chose D: This is incorrect. The formula \frac{dy}{dx} = ky \quad \text{or} \quad y'' + \omega^2 y = 0 is central to this subject. The other options are either simplified versions or unrelated.

Q3: What is the primary purpose of Partial Differential Equation (PDE)?

A) It is used to partial differential equation (pde) in mathematical analysis B) It is used only in advanced research contexts C) It replaces all other methods in this domain D) It is primarily a historical notation system

Correct: A)

• If you chose A: Partial Differential Equation (PDE) serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose B: This is incorrect. Partial Differential Equation (PDE) serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Partial Differential Equation (PDE) serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Partial Differential Equation (PDE) serves the purpose described in the correct answer. The other options misrepresent its role.

Q4: Which statement about Initial Value Problem (IVP) is TRUE?

A) Initial Value Problem (IVP) is an advanced topic beyond this subject's scope B) Initial Value Problem (IVP) is not related to this subject C) Initial Value Problem (IVP) is mentioned only as a historical footnote D) Initial Value Problem (IVP) is a fundamental concept covered in this subject

Correct: D)

• If you chose A: This is incorrect. Initial Value Problem (IVP) is a fundamental concept covered in this subject. This subject covers Initial Value Problem (IVP) as part of its core content.
• If you chose B: This is incorrect. Initial Value Problem (IVP) is a fundamental concept covered in this subject. This subject covers Initial Value Problem (IVP) as part of its core content.
• If you chose C: This is incorrect. Initial Value Problem (IVP) is a fundamental concept covered in this subject. This subject covers Initial Value Problem (IVP) as part of its core content.
• If you chose D: Initial Value Problem (IVP) is a fundamental concept covered in this subject. This subject covers Initial Value Problem (IVP) as part of its core content. Correct!

Q5: Based on the worked examples in this subject, what is the correct result?

A) A different result from a common mistake B) An unrelated numerical value C) The inverse of the correct answer D) Verify a Solution

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is Verify a Solution. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is Verify a Solution. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is Verify a Solution. The other options represent common errors.
• If you chose D: The worked examples show that the result is Verify a Solution. The other options represent common errors. Correct!

Q6: How are Initial Value Problem (IVP) and What Is A Differential Equation? related?

A) Initial Value Problem (IVP) and What Is A Differential Equation? are completely unrelated topics B) Initial Value Problem (IVP) is a special case of What Is A Differential Equation? C) Initial Value Problem (IVP) and What Is A Differential Equation? are closely related concepts D) Initial Value Problem (IVP) is the inverse of What Is A Differential Equation?

Correct: C)

• If you chose A: This is incorrect. Both Initial Value Problem (IVP) and What Is A Differential Equation? are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Initial Value Problem (IVP) and What Is A Differential Equation? are covered in this subject as interconnected topics.
• If you chose C: Both Initial Value Problem (IVP) and What Is A Differential Equation? are covered in this subject as interconnected topics. Correct!
• If you chose D: This is incorrect. Both Initial Value Problem (IVP) and What Is A Differential Equation? are covered in this subject as interconnected topics.

Q7: What is a common pitfall when working with Classification?

A) A common mistake is confusing Classification with a similar concept B) Classification is always computed the same way in all contexts C) The main error with Classification is using it when it is not needed D) Classification has no common misconceptions

Correct: A)

• If you chose A: Students often confuse Classification with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose B: This is incorrect. Students often confuse Classification with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Classification with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Classification with similar-sounding or related concepts. Pay attention to the precise definitions.

Q8: When should you apply Solutions?

A) Apply Solutions to solve problems in this subject's domain B) Solutions is not practically useful C) Use Solutions only in pure mathematics contexts D) Avoid Solutions unless explicitly instructed

Correct: A)

• If you chose A: Solutions is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose B: This is incorrect. Solutions is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Solutions is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Solutions is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Determine the order of $y^{(4)} - 3y'' + 2y = \sin x$.
   

   Click for answer

   Order 4 (highest derivative is $y^{(4)}$, the 4th derivative).

2. Is $yy' + y = x$ linear? Why or why not?
   

   Click for answer

   No — the term $yy'$ is a product of $y$ and its derivative. In a linear ODE, $y$ and its derivatives cannot be multiplied together.

3. Verify $y = x^2$ satisfies $xy' = 2y$.
   

   Click for answer

   $y = x^2$, $y' = 2x$. LHS: $x(2x) = 2x^2$. RHS: $2(x^2) = 2x^2$. ✓. In fact $y = Cx^2$ is the general solution.

4. How many arbitrary constants appear in the general solution of a 3rd-order ODE?
   

   Click for answer

   3 constants. An $n$-th order ODE has $n$ constants in its general solution.

5. Classify $y'' = \sin(y)$ (pendulum equation).
   

   Click for answer

   Second-order, nonlinear (because of $\sin y$), homogeneous, autonomous.

Summary

Key takeaways:

• ODE: single independent variable; PDE: multiple
• Classification: order (highest derivative), linearity, homogeneity, autonomy
• General solution contains $n$ arbitrary constants; IVP gives a particular solution
• Always verify solutions by plugging back into the DE
• Slope fields provide geometric insight

Pitfalls

• Confusing order with degree (or linearity): The "order" of an ODE is the highest derivative appearing, while "degree" (rarely used in modern texts) refers to the power on the highest derivative. More importantly, don't conflate "first-order" with "linear" — $y' = y^2$ is first-order but nonlinear. A linear ODE must have $y$ and its derivatives appearing only to the first power, not multiplied together.
• Thinking a general solution with $n$ constants covers all solutions: For an $n$-th order ODE, the general solution contains $n$ arbitrary constants — but this is only guaranteed for linear ODEs. Nonlinear ODEs can have singular solutions not obtainable from the general solution by choosing constants (e.g., $y = 0$ is a singular solution of $y' = y^{2/3}$ that the general solution family misses).
• Neglecting the existence and uniqueness conditions for IVPs: Not every IVP has a unique solution. The standard theorem requires continuity of $f$ and a Lipschitz condition on $f$ with respect to $y$. The classic counterexample $y' = \sqrt{y}, y(0) = 0$ has infinitely many solutions ($y=0$ for a while, then "peel off"). Always check whether the conditions hold before assuming a unique solution.
• Misclassifying homogeneous vs. nonhomogeneous when the RHS includes $y$ terms: A linear ODE is homogeneous if $g(x) = 0$ after all $y$ terms are moved to the left. The equation $y'' + 2y' = x$ is nonhomogeneous — but $y'' + 2y' = y$ (i.e., $y'' + 2y' - y = 0$) is homogeneous. The test is: can you write it as $a_n(x)y^{(n)} + \cdots + a_0(x)y = 0$?
• Interpreting slope fields incorrectly: Slope fields show direction at each point, but the line segments don't show magnitude — a long segment at a point doesn't mean the solution is fast there; segment length is just for visibility. Also, solution curves cannot cross except at equilibrium points or discontinuities — if it looks like they cross, you're probably misreading the field.

Next Steps

Next up: 27-02-first-order-odes.md — separable equations, integrating factors, and exact equations.