Non-Homogeneous & Constant-Coefficient ODEs: Root Classification, Euler Links, and Solution Building

Big-Picture Reminders

• Always learn every possible solution method.
  • One single substitution will not unlock every differential equation.
  • Instructor’s anecdote: a friend kept forcing one trick, succeeded three times, then stalled on the fourth problem – proof that breadth of techniques matters.
• Preview for the day
  • Transition from previously covered Bernoulli, homogeneous & linear cases to a new non-homogeneous, higher–order, constant–coefficient setting.
  • Strong emphasis on classifying roots (real, repeated, complex) and writing the correct form of the complementary solution $(y_c)$.

Anatomy of the Example Non-Homogeneous ODE

• Header in notes: “Between 21–24: non-homogeneous differential equation.”
• Generic layout: $a<em>2 y'' + a</em>1 y' + a_0 y = g(x)$
• Observations made in class:
  • First two terms ($y''$ and $y'$) have constant coefficients → no $x$ present, exponent “invisible zero.”
  • Last two terms involve $y$ and $x$, each carrying an exponent of $1$.
  • Presence of $x$ on the right (or any non-zero forcing $g(x)$) ⇒ equation is non-homogeneous.

Review: Powers of $i$ & Periodicity

• Key cycle: $i^0 = 1,\ i^1 = i,\ i^2 = -1,\ i^3 = -i,\ i^4 = 1 \text{ (cycle length 4)}$
• Fast exponent test: remainder mod $4$ decides the value.
  • Example:
  • $i^{17}$ → $17 \equiv 1 \, (\text{mod }4)$ → $i^{17} = i$.
  • $i^{35}$ → $35 \equiv 3 \, (\text{mod }4)$ → $i^{35} = -i$.

Euler’s Formula Refresher

• Fundamental identity: $e^{ix}=\cos x + i\sin x$.
• Immediate practice question: $e^{i\pi/2}=\cos(\pi/2)+i\sin(\pi/2)=i$.
• Derivative chain reminder: $\frac{d}{dx}\sin x = \cos x,$ and $\frac{d}{dx}\cos x = -\sin x$ follow from differentiating $e^{ix}$ term-by-term in its Maclaurin series (sometimes referred to as the “Euler function” in lecture).

Constant-Coefficient Homogeneous Theory

• For $n^{\text{th}}$-order ODE $a<em>n y^{(n)} + \dots + a</em>1 y' + a<em>0 y = 0$ form the characteristic polynomial $a</em>n r^n + \dots + a<em>1 r + a</em>0 = 0$.
• Three root categories and their solution pieces:
  • Distinct real roots $(r<em>1,\dots,r</em>k)$
  • $y<em>c = C</em>1 e^{r<em>1 x} + \dots + C</em>k e^{r_k x}$.
  • Repeated real root $r$ of multiplicity $m$
  • Contribution: $(C<em>1 + C</em>2 x + \dots + C_m x^{m-1}) e^{r x}$.
  • Complex conjugate pair $\alpha \pm i\beta$ (unrepeated)
  • Written once as $e^{\alpha x}(C<em>1 \cos \beta x + C</em>2 \sin \beta x)$.
  • Repeated complex pair: attach the same $x, x^2,\dots$ multipliers as in real-repeated case but wrapped around the sine/cosine block.
• Important vocabulary
  • $\alpha$ = real part, $\beta$ = imaginary part.

Worked Complex-Root Example ((2 \pm i))

1. Characteristic factor leads to roots $r = 2 \pm i$.
2. Write as $\alpha = 2,\ \beta = 1$.
3. Complementary solution: $y<em>c = e^{2x}(C</em>1 \cos x + C_2 \sin x)$.

Solving for Constants Using Initial Conditions

• Matrix viewpoint recommended (augmented matrix / row-reduction) for multiple $C_i$.
• Lecture mini-example (three unknowns):
  $\begin{cases}<br /> C<em>2 + C</em>3 &amp;= 7 \ <br /> 2C<em>2 - 5C</em>3 &amp;= 0 \ <br /> 4C<em>2 + 25C</em>3 &amp;= 70 <br /> \end{cases}$
• Student advised to review a short video on Gaussian elimination to speed up the algebra.

Rational Root Theorem Reminder for Factoring Characteristic Polynomials

• For integer coefficients, candidate rational roots are $\pm$ divisors of the constant term over divisors of the leading coefficient.
• Instructor’s quick list when constant term is $10$: ${\pm1,\pm2,\pm5,\pm10}$.
• Trial-and-error example: plug in $r=2$, find it zeros out the polynomial, then factor further.

Mixed Root Set Mega-Example (Book: p.170, Example 8)

• Eleven listed roots:
  $3,\ -5,\ 0 \text{ (multiplicity 4)},\ -5 \text{ (again)},\ 2+3i,\ 2-3i$
• Resulting complementary solution pieces:
  • $C_1 e^{3x}$ (distinct real)
  • $e^{-5x}(C<em>2 + C</em>3 x)$ (repeated real of mult. 2)
  • $e^{0x}(C<em>4 + C</em>5 x + C<em>6 x^2 + C</em>7 x^3) = C<em>4 + C</em>5 x + C<em>6 x^2 + C</em>7 x^3$ (zero root, mult. 4)
  • $e^{2x}(C<em>8 \cos 3x + C</em>9 \sin 3x)$ (complex pair)
• Instructor urges students to copy that full answer to “never forget” how each category appears.

Practical & Philosophical Takeaways

• Method diversity prevents dead-ends; know when each assumption (constant coefficients, linearity, etc.) is valid.
• Verify non-homogeneity before applying superposition/characteristic tricks.
• Imaginary solutions are expected for cubic and higher-degree polynomials → be comfortable switching to sine/cosine form.
• Augmented matrices accelerate the algebra of initial-value matching.
• Regular practice:
  • Evaluate $e^{i\theta}$ at common angles.
  • Compute $i^n$ for random $n$.
  • Factor polynomials with the Rational Root Theorem.
  • Re-write mixed-root solutions until the pattern is automatic.

Suggested Exercises & Resources

• Re-work the “between 21–24” problem counting non-zero forcing terms.
• Textbook p.170 Example 8 (11 roots) – copy solution structure verbatim, then derive it yourself.
• Watch a 5-minute crash video on Gaussian elimination / augmented matrices.
• Optional deep dive: derive $\sin$ and $\cos$ derivatives straight from the Maclaurin series of $e^{ix}$ to reinforce Euler’s formula.