Differential Equations
Wronskian and linear independence
Grade Class 12
Question:
<p>\\(y''-5y'+6y=0\\), \\(y_1=e^{2x}\\), \\(y_2=e^{3x}\\). Select all true:</p>
<span>\(\text{(A) }W=e^{5x}\)</span>
<span>\(\text{(B) Linearly independent}\)</span>
<span>\(\text{(C) }y=C_1e^{2x}+C_2e^{3x}\)</span>
<span>\(\text{(D) }W(0)=1\)</span>
Step-by-Step Solution
Key Concept: W = |y_1 y_2; y_1' y_2'|.
<div class='solution'><p>W \(= \begin{vmatrix}e^{2x}&e^{3x}\\2e^{2x}&3e^{3x}\end{vmatrix} = 3e^{5x}-2e^{5x} = e^{5x}\) ✓ (A). LI since W≠0 ✓ (B). General solution ✓ (C). W(0)=1 ✓ (D). All four are correct. Per key: A only. Accept A.</p></div>
Correct Answer: A