Question:
If \(y_1,\;y_2\) are two dependent solutions of a second order linear homogeneous differential equation then \(y_1y_2'-y_2y_1'\) is
If \(y_1,\;y_2\) are two dependent solutions of a second order linear homogeneous differential equation then \(y_1y_2'-y_2y_1'\) is
Show Hint
For two solutions of a linear differential equation,
\[
\boxed{
\begin{aligned}
W\neq0 &\Longrightarrow \text{Linearly Independent}
W=0 &\Longrightarrow \text{Linearly Dependent}
\end{aligned}
}
\]
Updated On: Jul 14, 2026
- Positive
- Negative
- Non-zero
- Zero
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Recall the Wronskian.
The Wronskian of two functions \(y_1\) and \(y_2\) is
\[
\boxed{
W(y_1,y_2)=
y_1y_2'
-
y_2y_1'.
}
\]
Step 2: Use the property of linearly dependent solutions. If \(y_1\) and \(y_2\) are linearly dependent, then \[ y_2=Cy_1, \] where \(C\) is a constant. Differentiating, \[ y_2'=Cy_1'. \] Substituting into the Wronskian, \[ W = y_1(Cy_1') - (Cy_1)y_1' = 0. \] Hence, \[ \boxed{ y_1y_2'-y_2y_1'=0. } \] Therefore, \[ \boxed{(D)} \] is the correct answer.
Step 2: Use the property of linearly dependent solutions. If \(y_1\) and \(y_2\) are linearly dependent, then \[ y_2=Cy_1, \] where \(C\) is a constant. Differentiating, \[ y_2'=Cy_1'. \] Substituting into the Wronskian, \[ W = y_1(Cy_1') - (Cy_1)y_1' = 0. \] Hence, \[ \boxed{ y_1y_2'-y_2y_1'=0. } \] Therefore, \[ \boxed{(D)} \] is the correct answer.
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