Subtopic: Higher order linear ODEs with constant coefficients
Question Statement:
Consider the second-order homogeneous linear differential equation $y'' + P(x) y' + Q(x) y = 0$ with two solutions $y_1(x)$ and $y_2(x)$ on an interval $I$. Which of the following statements is/are TRUE?
Options:
(A) The Wronskian of the two solutions is defined as $W(y_1, y_2) = y_1 y_2' - y_1' y_2$
(B) $y_1$ and $y_2$ form a fundamental basis of solutions if and only if $W(y_1, y_2) \neq 0$ everywhere on $I$
(C) Abel's identity states that $W(x) = W(x_0) \exp\left( -\int_{x_0}^x P(t)\,dt \right)$
(D) If $W(x_0) = 0$ at any single point $x_0 \in I$, then $W(x) = 0$ for all $x \in I$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. **Wronskian Determinant (Option A)**: $W = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_1' y_2$ (True).
2. **Linear Independence Criterion (Option B)**: Two solutions are linearly independent on $I$ if and only if their Wronskian is non-zero everywhere on $I$ (True).
3. **Abel's Identity (Option C)**: Differentiating $W' = y_1 y_2'' - y_1'' y_2 = -P(x) W$ yields the exponential relation $W(x) = W(x_0) e^{-\int_{x_0}^x P(t)\,dt}$ (True).
4. **All-or-Nothing Property (Option D)**: Because the exponential function never vanishes, $W(x)$ is either identically zero or nowhere zero on $I$ (True).