DiffEq test 2 Equations

Auxiliary Equation (Constant Coeffs)

ar^2 + br + c = 0 for the ODE ay'' + by' + c*y = 0.

Distinct Real Roots (Case 1)

y(x) = c1e^(r1x) + c2e^(r2x) (where r1 != r2).

Repeated Real Root (Case 2)

y(x) = c1e^(rx) + c2x*e^(rx) (multiplied by x for linear independence).

Derivative of Repeated Root Term (c2xe^(rx))

Use Product Rule: d/dx[c2xe^(rx)] = c2e^(rx) + c2rxe^(rx). At x = 0, this equals c2.

Complex Conjugate Roots (Case 3: r = alpha +/- i*beta)

y(x) = e^(alpha*x) * [c1cos(betax) + c2sin(betax)].

Pure Imaginary Roots (alpha = 0)

y(x) = c1cos(betax) + c2sin(betax) (since e^(0x) = 1).

Derivative of Pure Trig: d/dx[c1cos(kx) + c2sin(kx)]

Chain Rule: -kc1sin(kx) + kc2cos(kx).

Mass-Spring Governing Equation

my'' + by' + k*y = 0 (m = mass, b = damping coefficient, k = spring constant).

Damping Discriminant Formula

Delta = b^2 - 4mk.

Undamped Classification Condition

b = 0 (pure imaginary roots r = +/- i*omega; oscillates indefinitely).

Underdamped Classification Condition

Delta = b^2 - 4mk < 0 (complex roots with negative real part; decaying oscillations).

Critically Damped Classification Condition

Delta = b^2 - 4mk = 0 (repeated negative real root; returns to equilibrium fastest without oscillating).

Overdamped Classification Condition

Delta = b^2 - 4mk > 0 (distinct negative real roots; sluggish return to equilibrium without oscillating).

Natural Angular Frequency (omega)

omega = sqrt(k / m) [in rad/s].

Period (T) in Undamped Motion

T = 2pi / omega = 2pi * sqrt(m / k).

Amplitude (A) in Undamped Motion

A = sqrt(c1^2 + c2^2) for y(t) = c1cos(omegat) + c2sin(omegat).

Mass-Spring Sign Conventions

Down / Right = positive (+); Up / Left = negative (-).

Full Nonhomogeneous General Solution Structure

y(x) = y_c(x) + y_p(x) (Complementary homogeneous + Particular solution).

Superposition Principle for Particular Solutions

If L[yp1] = g1(x) and L[yp2] = g2(x), then L[k1yp1 + k2yp2] = k1g1(x) + k2g2(x).

Undetermined Coefficients: Guess for Polynomial (Pn(x))

yp = Anx^n + … + A1x + A0 (include all lower-degree terms down to constant).

Undetermined Coefficients: Guess for Exponential (e^(kx))

yp = A*e^(kx).

Undetermined Coefficients: Guess for Trig (cos(kx) or sin(kx))

yp = Acos(kx) + Bsin(kx) (always include BOTH sin and cos).

Undetermined Coefficients: Overlap / Multiplication Rule

If any term in the trial yp overlaps with yc(x), multiply the entire block by x (or x^2 if repeated) until no duplicates remain.

Cauchy-Euler Standard Form

ax^2y'' + bxy' + c*y = 0 (power of x matches derivative order, x > 0).

Cauchy-Euler Auxiliary Equation

am(m - 1) + bm + c = 0 ==> am^2 + (b - a)m + c = 0.

Cauchy-Euler: Common Exam Trap in Auxiliary Equation

The middle coefficient is (b - a), NOT just b!

Cauchy-Euler: Distinct Real Roots (Case 1)

y(x) = c1x^(m1) + c2x^(m2).

Cauchy-Euler: Repeated Real Root (Case 2: m1 = m2 = m)

y(x) = c1x^m + c2x^m*ln(x) (multiplied by ln(x), NOT x!).

Cauchy-Euler: Complex Conjugate Roots (Case 3: m = alpha +/- i*beta)

y(x) = x^alpha * [c1cos(betaln(x)) + c2sin(betaln(x))].