Calc 2-2

Midterm 2

Second Order Homogeneous Linear Differential Equations (DEs)

• Definition: Second-order linear differential equations have the general form:

  • $\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = G(x)$
  • Parameters: Where $P, Q, R,$ and $G$ are continuous functions of $x$ on a given interval.

• Sub-Classification: Second-order homogeneous linear differential equations specifically have the form:

  • $\frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + R(x)y = 0$
  • Necessity: Here, $P, Q,$ and $R$ are still continuous functions of $x$ on a given interval.

• Sub-Sub-Classification: The second-order homogeneous linear differential equations with constant coefficients are expressed as:

  • $ay'' + by' + cy = 0$
  • Parameters: Where $a, b,$ and $c$ are constants.

Roots of Characteristic Equation

• The general characteristic equation derived from the constant coefficient version is given by:
  • $ext{Characteristic: } ar^2 + br + c = 0$
  • Roots:
  • Real and Distinct Roots: If the discriminant D = b^2 - 4ac > 0, then two real roots exist.
  • Repeated Roots: If $D = 0$, there is one repeated real root.
  • Complex Roots: If D < 0, roots are complex conjugates.

Example Differential Equation

• An example of a second-order homogeneous linear differential equation:
  • $y'' - Sy' + 2y = 0$

Second Order Nonhomogeneous Linear Differential Equations

• Form: The general form of a second-order nonhomogeneous linear differential equation is:

  • $ay'' + by' + cy = G(x)$
  • Characteristics: Here, $a, b,$ and $c$ continue to be constants while $G$ remains a continuous function.

• Complementary (Homogeneous) Equation

  • Its corresponding homogeneous equation is:
  • $ay'' + by' + cy = 0$

• General Solution: The solution to the nonhomogeneous DE can be expressed as:

  • $y(x) = Y_p(x) + Y_c(x)$
  • Where $Y_p$ is a particular solution to the nonhomogeneous DE, and $Y_c$ is the general solution of the complementary equation.

Determining Particular Solutions (Method of Undetermined Coefficients)

• Process:
  • If $G(x)$ is a polynomial of degree $k$, assume $Y_p(x)$ is also a polynomial of degree $k$.
  • If $G(x)$ is $ext{sin}(kx)$ or $ext{cos}(kx)$, assume:
  • $Y_p(t)$ to be a combination of both trigonometric functions.

Deriving Specific Examples

• For example, let the particular solution