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If y1 and y2 are two solutions o f y" + p{x)y' + q(x)y - 0, then for general solution of this given equation, y1 and y2 are
- a)linearly dependent
- b)linearly independent
- c)proportional
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If y1 and y2 are two solutions o f y" + p{x)y + q(x)y - 0, then f...
Linear Dependence of Solutions
When it comes to the solutions of a second-order linear homogeneous differential equation, it is important to analyze whether the solutions are linearly dependent or independent.
Definition of Linear Dependence
Two functions are said to be linearly dependent if one of them can be expressed as a constant multiple of the other. In this context, if y1 and y2 are solutions of the differential equation, they are linearly dependent if one can be written as a constant multiple of the other.
Explanation
- If y1 and y2 are solutions of the given differential equation, they are considered linearly independent if no constant multiple of one function can produce the other function.
- In other words, if y1 and y2 are linearly independent solutions, they cannot be written as a constant multiple of each other.
- This property is crucial in finding the general solution of the differential equation, as linearly independent solutions provide a basis for the solution space.
- If the solutions were linearly dependent, the general solution would not be as expressive and would not capture the full range of possible solutions.
Conclusion
In the case of the given differential equation, if y1 and y2 are two solutions, the general solution will involve a linear combination of these solutions. Therefore, the solutions y1 and y2 are linearly independent, as they form a basis for the general solution space.
When it comes to the solutions of a second-order linear homogeneous differential equation, it is important to analyze whether the solutions are linearly dependent or independent.
Definition of Linear Dependence
Two functions are said to be linearly dependent if one of them can be expressed as a constant multiple of the other. In this context, if y1 and y2 are solutions of the differential equation, they are linearly dependent if one can be written as a constant multiple of the other.
Explanation
- If y1 and y2 are solutions of the given differential equation, they are considered linearly independent if no constant multiple of one function can produce the other function.
- In other words, if y1 and y2 are linearly independent solutions, they cannot be written as a constant multiple of each other.
- This property is crucial in finding the general solution of the differential equation, as linearly independent solutions provide a basis for the solution space.
- If the solutions were linearly dependent, the general solution would not be as expressive and would not capture the full range of possible solutions.
Conclusion
In the case of the given differential equation, if y1 and y2 are two solutions, the general solution will involve a linear combination of these solutions. Therefore, the solutions y1 and y2 are linearly independent, as they form a basis for the general solution space.
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