MA 3253 (Differential Equation I) Lecture 10
Section 4.3
ficients
Homogeneous Linear Equations with Constant Coef-
Goal: In this section, we study how to solve for
ay 00 + by 0 + cy = 0,
where a, b, c are constants.
Assume that y = emx is a solution of (1),
Auxiliary Equation
The auxiliary equation of the differential equation (1) is
The two roots are
There will be three cases for the roots.
• m1 and m2 are real and distinct,
• m1 and m2 are real and equal,
• m1 and m2 are a pair of complex conjugate numbers,
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(1)
Case I: Dinstinct Real Roots
If the auxiliary equation has two real distinct roots m1 and m2 , then
The general solution is
Example 1. Solve the differential equation
2y 00 − 5y 0 − 3y = 0.
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Case II: Repeated Real Roots
If m1 = m2 , then
We need to find another solution to the differential equation.
Example 2. Solve the differential equation
y 00 − 10y 0 + 25y = 0.
In general, if the auxiliary equation has two repeated real roots, m1 = m2 , then
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Case III: Conjugated Complex Roots
If m1 and m2 are complex, then we can write
Euler’s formula:
Instead of using y1 = e(α+iβ)x , y2 = e(α−iβ)x to form the general solutions,
Example 3. Solve the differential equation
y 00 + 4y 0 + 7y = 0.
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Higher Order Equations
Example 4. Solve the differential equation
y 000 + 3y 00 − 4y = 0.
Example 5. Solve the differential equation
y (4) + 2y 00 + y = 0.
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