Calc 2 - 11/21

Linear Differential Equations Overview

• Linear differential equations have the property that any linear combination of solutions is also a solution.
• An equation is homogeneous if set equal to zero; non-homogeneous if there is a function on the right-hand side.
• Recognized as second order if the second derivative appears.

Characteristics of Linear Equations

• Only linear combinations of solutions are valid (e.g., no squared or trigonometric terms of the dependent variable).
• Non-linear terms like $ext{sine}$ or $ext{cosine}$ of the dependent variable indicate the equation is non-linear.

Determining Linear Equations

• An equation is linear if derivative terms (e.g., $y', y''$) are not squared or contain non-linear functions of $y$.
• For second order equations:
  • $y'' = -4y'^{2}$ is not linear due to $y'^{2}$.
  • Including functions of $x$ (e.g., $x^{2}$) is permissible if dependent variables maintain linearity.

Solving Linear Homogeneous Equations

• Solution technique involves substitution of $y = e^{rx}$ to derive the characteristic equation.
• Characteristic equation yields three cases based on the discriminant:
  1. Two distinct real roots (b^{2} - 4ac > 0)
  2. Repeated root ($b^{2} - 4ac = 0$)
  3. Complex roots (b^{2} - 4ac < 0)

Case Breakdown

1. Distinct Roots: Solution form: $y<em>h = c</em>1 e^{r<em>1 x} + c</em>2 e^{r_2 x}$
2. Repeated Root: Solution form: $y<em>h = c</em>1 e^{rx} + c_2 x e^{rx}$
3. Complex Roots: Solutions involve trigonometric functions.

General Solution Format

• General form utilizes character roots derived from the characteristic equation.
• Example: For the equation $y'' - 4y' + 4y = 0$, use $c<em>1 e^{2x} + c</em>2 x e^{2x}$ for repeated roots.

Finding Constants

• Initial conditions are required to solve for constants $c<em>1$ and $c</em>2$.
• Derivative terms must be managed using the product rule if necessary when applying initial conditions.