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Note

Constant-Coefficient Homogeneous Equations

Definition

A constant-coefficient homogeneous equation is a second-order linear ODE where the coefficients a,b,ca, b, c are real constants: ay′′+by′+cy=0,a≠0ay'' + by' + cy = 0, \quad a \neq 0

  • How to read: “The expression a y double prime plus b y prime plus c y equals zero, where a is not zero.”
  • Meaning: Second-order linear homogeneous ODE with constant coefficients—the characteristic-equation method applies.

Why It Matters

They describe the core behavior of vibrating systems and electrical circuits, allowing engineers to predict oscillations and damping.

Core Concepts

  • Characteristic Equation

    • Assume solution y=erxy = e^{rx} → plug in to get ar2+br+c=0ar^2 + br + c = 0.
    • How to read: “The expression a r squared plus b r plus c equals zero.”
    • Meaning: Substituting y=erxy = e^{rx} reduces the ODE to this quadratic; solve with the quadratic formula r=−b±b2−4ac2ar = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

  • Case I: Distinct Real Roots (b² - 4ac > 0) y=c1er1x+c2er2xy = c_1 e^{r_1 x} + c_2 e^{r_2 x}

    • How to read: “The solution y equals c one times e to the power r one x plus c two times e to the power r two x.”
    • Meaning: Two distinct real roots r1,r2r_1, r_2 give a general solution as a linear combination of exponentials.

  • Case II: Repeated Real Root (b² - 4ac = 0) y=(c1+c2x)erxy = (c_1 + c_2 x) e^{r x}

    • How to read: “The solution y equals the quantity c one plus c two x times e to the power r x.”
    • Meaning: Repeated root rr requires the extra xx factor (from reduction of order) for a second independent solution.

  • Case III: Complex Roots (b² - 4ac < 0) Let r = α ± i β. Then: y=eαx(c1cos⁡(βx)+c2sin⁡(βx))y = e^{\alpha x} (c_1 \cos(\beta x) + c_2 \sin(\beta x))

    • How to read: “The solution y equals e to the power alpha x times the quantity c one cosine beta x plus c two sine beta x.”
    • Meaning: Complex roots α±iβ\alpha \pm i\beta yield oscillatory solutions with exponential envelope eαxe^{\alpha x}.

These three cases cover all possibilities for the second-order linear homogeneous constant-coefficient ODE. The form of the solution is completely determined by the nature of the roots of the characteristic polynomial.

Connected Concepts

Connected notes