Find the First Six Terms of the Power Series Representing $\left( x ^ { 2 } + 2 \right) y ^ { tt } + y = 0$

question 6

Multiple Choice

Find the first six terms of the power series representing independent solutions of the differential equation $\left( x ^ { 2 } + 2 \right) y ^ { tt } + y = 0$ .

• A)\(y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } - \frac { x ^ { 4 } } { 32 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } - \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)\), where \(a _ { 0 }\) and \(a _ { 1 }\) are arbitrary constants
• B) \(y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 96 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)\), where \(a _ { 0 }\) and \(a _ { 1 }\) are arbitrary constants
• C) \(y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 32 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)\), where \(a _ { 0 }\) and \(a _ { 1 }\) are arbitrary constants
• D) \(y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 8 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 6 } + \frac { 7 x ^ { 5 } } { 120 } - \cdots \right)\), where \(a _ { 0 }\) and \(a _ { 1 }\) are arbitrary constants
• E) \(y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 16 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 240 } - \cdots \right)\), where \(a _ { 0 }\) and \(a _ { 1 }\) are arbitrary constants

Understand the role of pathogen Avr genes in eliciting plant defense responses.

Recognize the role of secondary metabolites in plant defense against herbivory.

Explain the importance of physical barriers and hypersensitive responses in plant defense.

Understand the evolution and ecological significance of plant defense strategies.

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