Subject test
Differential equation whose linearly independent solutions are, cos 2x, sin2x ande^-x
(a) (D'+D? + 4D + 4) y = 0)
(b) (D? - D’ +4D - 4) y = 0
(c) (D? + D² - 4D - 4) y = 0
(d) (D? - D? - 4D + 4) y = 0
Answers
Step-by-step explanation:
i thing no b is the ans ok
Concept-
Use the concept of differential equations to solve this question. A differential equation is an equation that contains one or more terms and the derivatives of one variable with respect to another variable.
Given-
Differential equations whose linearly independent solutions are given as cos 2x , sin 2x , e ^x.
Find-
Find the correct differential equation from the options given ,whose linearly independent solutions are cos 2x, sin 2x , e ^x.
Solution-
( D³ - D² + 4D - 4 )y = 0
[ D² ( D - 1 ) + 4 ( D - 1)y = 0
( D²+ 4 ) ( D - 1 )y = 0
(D - 1 )y = 0 or ( D² + 4 )y = 0
For (D-1)y = 0 ,
Take , y = e ^mx
( D - 1 ) e ^mx = 0
⇒ (m - 1 ) e ^mx = 0
⇒ m = 1 (∵ e ^mx ≠ 0)
One solution, y = e ^x
For ( D² + 4 )e ^mx = 0
⇒ ( m² + 4 ) e ^mx = 0
⇒ m² + 4 = 0 ( ∵ e ^mx ≠ 0 )
⇒ m² = -1 × 4
⇒ m = ± √-1 √4 = ± 2i
⇒ y = Ae ^2ix + Be ^-2ix
which can be written as ⇒ C cos 2x + D sin 2x
[ ∵ cos 2x = e ^2ix + e^-2ix / 2 ] and
[ ∵ sin 2x = e ^2ix - e ^-2ix / 2 ]
Therefore, for option B , we get cos 2x , sin 2x , e ^x as linearly independent solution.
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