2nd Order Homogeneous Linear Differential Equation
General Form
Auxiliary Equation
The roots of the Auxiliary Equation (or Characteristic Equation) can be found by Factoring.
In other cases, the roots are found by using the quadratic formula.
where
is called the discriminant.
the roots are real and unequal
the roots are real and equal
the roots are complex numbers
Determine the General Solution of the 2nd Order Homogeneous Linear Differential Equation.
Determine the Solution of the 2nd Order Homogeneous Linear Differential Equation with Initial Value.
Answers to Odd Number questions.
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2nd Order Non-Homogeneous Linear Differential Equation
General Form
Case I:
G(t)
where G(t) is a polynomial function
Case II:
G(t)
where G(t) is an exponential function
Case III:
G(t)
where G(t) is a cosine/sine function
When
G(t)
is a sum of several functions, solve the particular solutions individually and then add.
exponential function
polynomial function
cosine/sine function
Important:
1. Note that Y(t) or the particular solution is only for G(t).
2. Still need to determine the homogeneous solution or the complimentary solution.
Final Solution
Determine the General Solution of the 2nd Order Non-Homogeneous Linear Differential Equation.
Determine the Solution of the 2nd Order Non-Homogeneous Linear DE with Initial Value.
Answers to questions.