3.1 Theory of Linear Equations

Assigned HW: 4, 7, 8, 9, 10, 16, 17, 21, 23, 25, 34, 36.

A general linear first-order differential equation is $P(t) \frac{dy}{dt} + Q(t)y = R(t)$.

• Existence of a unique solution

  

• Linear Dependence/Independence

  • A set of functions is linearly independent on an interval if no non-trivial combination of these functions equals zero. If such a combination exists, then the functions are linearly dependent.

    

• Wronskian: used to test whether two functions are linearly independent

  $$ W(y_1, y_2)(t) = \det\begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} $$

  • If the Wronksian is nonzero at any point of the interval, then the solutions are linearly independent on that interval

• Criterion for Linearly Independent Solutions

  

  • If you find your set of functions is linearly independent on the interval AND they are also solutions to the differential equation on that same interval, then they form a fundamental set of solutions

• Fundamental Set of Solutions: refers to a set of solutions that are linearly independent and span the solution space (every solution to the DE can be expressed as a linear combination of the solutions in the fundamental set)

  

• Existence of a Fundamental Set

  

• Homogenous Equations

  

• Non-homogenous Equations

  

  • $y = y_h + y_p$
  • $y_h = c_1y_1+c_2y_2 + \dots$
  • $y_1$ and $y_2$ are linearly independent solutions

• Complementary Function: the general solution of the associated homogenous equation

  

3.2 Reduction of Order

Assigned HW: 7, 9, 12, 15, 20, 24.

Used when you have one solution $y_1(x)$ to the homogenous equation and are looking for a second linearly independent solution

3.3 Linear Equations with Constant Coefficients

Assigned HW: 7, 10, 11, 17, 19, 33, 39, 44, 45, 47, 49, 52.

1. Case I: Distinct Real Roots