Math, asked by ckanti237, 7 months ago

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

Answered by mmaaa4460
3

Step-by-step explanation:

i thing no b is the ans ok

Answered by arshikhan8123
0

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.

#SPJ2

Similar questions