Basic Concepts
Moving up a level. Second derivatives ($y''$) allow us to model acceleration, forces, and vibrations.
Introduction
A Second Order Linear Differential Equation has the form:
Most of the time, we divide by $P(t)$ to get the standard form:
If $g(t) = 0$, the equation is called Homogeneous. Otherwise, it is Nonhomogeneous.
Principle of Superposition
Theorem
If $y_1(t)$ and $y_2(t)$ are two solutions to a Homogeneous linear equation, then any linear combination is also a solution:
To represent all possible solutions, $y_1$ and $y_2$ must be "different enough" (Linearly Independent). We check this using the Wronskian:
.
Visual: Linear Combinations
Interactive: Mixing Solutions
For , the functions and are basic building blocks. The general solution is a weighted sum.
Worked Examples
Example 1: Verifying Solutions
Verify that and are solutions to for .
1. Check :
.
Plug in: . Verified.
2. Check :
.
Plug in: . Verified.
3. General Solution:
.
Example 2: Initial Value Problem
Check that solves . Find constants given .
1. Verify:
If , . So works.
2. Apply :
. So .
3. Apply :
.
.
4. Solution:
.
Example 3: Wronskian
Are and linearly independent solutions to ?
1. Calculate Wronskian:
.
.
2. Conclusion:
Since , the solutions are linearly independent and form a fundamental set.
Practice Quiz
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