Warm-up Problem The function y1 = ex/2 is a solution to the differential equation 4y – 4y + y = 0. Use reduction of order to find a second solution.

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Presentation transcript:

Warm-up Problem The function y1 = ex/2 is a solution to the differential equation 4y – 4y + y = 0. Use reduction of order to find a second solution.

Homog. Equations with Constant Coefficients Consider ay + by + cy = 0. A solution will have the form y = emx This is called the auxiliary equation.

3 Cases If distinct roots m1 and m2  If repeating root m  If complex conjugate roots α ± βi 

Ex. Solve 2y – 5y – 3y = 0

Ex. Solve y – 10y + 25y = 0

Ex. Solve y + 4y + 7y = 0

Ex. Solve the IVP 4y + 4y + 17y = 0, y(0) = –1, y(0) = 2

y + k2y = 0 has the solution Useful to remember: y + k2y = 0 has the solution

This works for higher order equations, but it’s harder to break down the cases: If all roots are distinct m1, m2, …, mn  If a root m repeats k times 

Remember, if there is a complex root, then its conjugate is also a root: α ± βi  A cubic polynomial must have at least one real root.

Ex. Solve y + 3y – 4y = 0

Ex. Solve y + 2y + y = 0

Ex. Solve y – 3y = 0