MATH102

Definition: First-Order Linear Differential Equation

An equation of the form dy/dx + P(x)y = Q(x).

Method: Integrating Factor

The function I(x) = exp(∫ P(x) dx) used to solve first-order linear differential equations.

Definition: Separable Differential Equation

An equation that can be written in the form dy/dx = g(x)h(y), solved by integrating 1/h(y) dy = g(x) dx.

Definition: Second-Order Linear Homogeneous Equation

An equation of the form a(d²y/dx²) + b(dy/dx) + cy = 0, where a, b, and c are constants.

Definition: Characteristic Equation

The quadratic equation am² + bm + c = 0 used to find the general solution of a second-order linear homogeneous ODE.

General Solution: Distinct Real Roots (m1 ≠ m2)

y(x) = Ae^{m1x} + Be^{m2x}

General Solution: Repeated Real Roots (m1 = m2 = m)

y(x) = (A + Bx)e^{mx}

General Solution: Complex Roots (m = α ± iβ)

y(x) = e^{αx}[A cos(βx) + B sin(βx)]

Definition: Partial Derivative

The derivative of a multivariable function with respect to one variable, keeping all other variables constant.

Definition: Clairaut's Theorem (Mixed Partials)

If the partial derivatives f_xy and f_yx are continuous on an open set, then f_xy = f_yx.

Definition: Gradient Vector (∇f)

The vector of all first-order partial derivatives of a multivariable function: ∇f = (∂f/∂x, ∂f/∂y, …).

Definition: Directional Derivative

The rate of change of f in the direction of a unit vector u, calculated as D_u f = ∇f · u.

Definition: Jacobian Matrix

The matrix of all first-order partial derivatives of a vector-valued function.

Definition: Double Integral over a Rectangle

The volume under the surface z = f(x,y) over a rectangular region, computed as ∫{a}^{b} ∫{c}^{d} f(x,y) dy dx.

Theorem: Fubini's Theorem

If f(x,y) is continuous on a rectangular region R, the order of integration in a double integral does not matter.

Polar Coordinates Transformation

x = r cos(θ), y = r sin(θ), and the differential area element dA = r dr dθ.

Definition: Infinite Series Convergence

A series ∑ a_n converges if the sequence of its partial sums S_k = ∑_{n=1}^{k} a_n approaches a finite limit as k → ∞.

Test: Geometric Series Ratio Test

A geometric series ∑ c*r^n converges if and only if |r| < 1, and its sum is a / (1 - r).

Test: The Divergence Test

If lim_{n→∞} a_n ≠ 0 (or does not exist), then the infinite series ∑ a_n must diverge.

Test: Alternating Series Test (Leibniz's Theorem)

An alternating series ∑ (-1)^n b_n converges if b_n is monotonically decreasing and lim_{n→∞} b_n = 0.

Definition: Taylor Series of f(x) centered at c

∑_{n=0}^{∞} [f^(n)(c) / n!] * (x - c)^n

Definition: Maclaurin Series

A Taylor series centered specifically at the point c = 0.

Maclaurin Series: e^x

∑_{n=0}^{∞} (x^n / n!) = 1 + x + x²/2! + x³/3! + …

Maclaurin Series: sin(x)

∑_{n=0}^{∞} [(-1)^n * x^(2n+1)] / (2n+1)! = x - x³/3! + x⁵/5! - …

Maclaurin Series: cos(x)

∑_{n=0}^{∞} [(-1)^n * x^(2n)] / (2n)! = 1 - x²/2! + x⁴/4! - …

Definition: Laplace Transform

An integral transform that converts a function of time t into a function of complex frequency s: L{f(t)} = ∫_{0}^{∞} e^{-st}f(t)dt.