June 15, 2026 - Calculus 2 - Introduction to Differential Equations and Numerical Methods (Concise)

Verifying Solutions to Differential Equations

• A differential equation involves a function $y$ and one or more of its derivatives.

• To verify a possible solution, calculate the required derivatives and substitute them into the original equation.

• Example: To check a solution for a second-order equation involving $y$, $y'$, and $y''$, take two derivatives of the function (using rules like the chain rule for expressions such as $e^{(2x)}$) and simplify the resulting expression to ensure it satisfies the given equality.

Ordering and Notation

• Order: The order of a differential equation is determined by the highest order derivative present in the equation.

  • First order: Contains only the first derivative ($y'$).

  • Second order: Contains the second derivative ($y''$).

  • Third order: Contains the third derivative ($y'''$).

• High-Order Notation: For derivatives higher than the third order, numerical subscripts in parentheses are used (e.g., $y^{(4)}$ or $y^{(10)}$) to avoid excessive apostrophes and clarify the notation from powers like $y^{4}$.

General and Particular Solutions

• General Solution: Represents a family of solutions and typically includes an arbitrary constant $C$. For example, for $y' = 2 \times x$, the general solution is $y = x^{2} + C$.

• Particular Solution: A specific solution where the constant $C$ is determined based on a specific point the function passes through, such as $(2, 7)$.

• Initial Value Problem (IVP): Refers to a differential equation accompanied by an initial condition (e.g., $y(0) = 3$). These conditions allow for finding the unique particular solution that models a specific situation, often starting at time $t = 0$.

Physics Applications

• Differential equations model real-world behaviors by relating functions to their derivatives.

• Kinematics:

  • Acceleration ($a$) is the derivative of velocity ($v'$).

  • Velocity ($v$) is the derivative of position ($s'$).

• Example Case: A baseball with mass $0.15\,kg$ is thrown upward with an initial velocity of $10\,m/s$ from a height of $3\,m$.

  • Acceleration due to gravity is $-9.8\,m/s^{2}$.

  • $v(t) = \int -9.8\,dt = -9.8 \times t + C$. Using initial velocity $v(0) = 10$, the particular solution is $v(t) = -9.8 \times t + 10$.

  • $s(t) = \int (-9.8 \times t + 10)\,dt = -4.9 \times t^{2} + 10 \times t + C$. Using initial position $s(0) = 3$, the particular solution is $s(t) = -4.9 \times t^{2} + 10 \times t + 3$.

Direction Fields and Equilibrium Solutions

• Direction Field (Slope Field): A visual tool where arrows represent the slope of the tangent line ($y'$) at various points in the plane. This provides a qualitative view of how solutions behave even if they cannot be solved analytically.

• Equilibrium Solutions: These occur when the derivative is zero ($y' = 0$). These solutions represent horizontal lines where the function value remains constant over time.

• Stability: If tangent lines push toward an equilibrium solution, the function stabilizes; if they push away, it is unstable.

Numerical Methods: Euler's Method

• Concept: Used when an anti-derivative is difficult or impossible to find. It employs linear approximation to estimate values of a function step-by-step.

• Linearization Formula: $L(x) = f'(a)(x - a) + f(a)$.

• Implementation:

  • Start at a known point $(a, f(a))$.

  • Estimate the next point by moving a small step size ($h$) in the direction of the local slope.

  • Use the new point to rebuild a new linearization and repeat the process.

• Accuracy: Smaller step sizes yield better approximations, but because each step is an estimation based on a previous estimation, errors can accumulate.

Questions & Discussion

Question regarding Order Notation: How do you distinguish a higher-order derivative from a power, such as $y^{(4)}$ vs. $y^{4}$? Response: Context is key in differential equations. Notationally, the use of parentheses around the exponent indicates a derivative. Additionally, taking a non-integer derivative (like a half-derivative) is not standard in this context.

Question regarding Equilibrium Solutions: Why is a vertical line like $x = 3$ not considered a solution to the differential equation? Response: A vertical line does not pass the vertical line test and therefore cannot be considered a function $y = f(x)$. Differential equation solutions must be functions.

Question regarding Numerical Accuracy: If you use an estimation to make a further estimation, does it get more wrong? Response: Yes, it is an estimation of an estimation, meaning the error can drift further from the true curve the longer the process continues. This is why small steps are necessary for better results.