Question

solve
the
Differential
equation.
(D² - 4D + 4)y =
sin2x​

Answer 1

Answer :-

$\to \boxed{\sf{\frac{ {d}^{2} y}{d {x}^{2} } + 4y = 0}} \:$

Explanation :-

We have

$\to \sf{({D}^{2}- 4D + 4 \: )y = \sin2x \: \: ......eq.(1)} \\ \\ \sf{differentiate \: with \: respect \: to \: x \: } \\ \\ \to \sf{ \frac{dy}{dx} ({D}^{2}- 4D + 4 ) = 2 \cos2x} \\ \\ \sf{ again \: differentiate \: with \: respect \: to \: x \: } \: \\ \\ \to \sf{\frac{ {d}^{2} y}{d {x}^{2} } ({D}^{2}- 4D + 4 ) = - 4 \sin2x}$

from eq.(1)

$\to \sf{ \frac{ {d}^{2}y }{d {x}^{2} }({D}^{2}- 4D + 4 ) = - 4({D}^{2}- 4D + 4 )y \: } \\ \\ \to \sf{ \frac{ {d}^{2}y }{ d{x}^{2} } = - 4y} \\ \\ \to \boxed{\sf{\frac{ {d}^{2} y}{d {x}^{2} } + 4y = 0}}$

this is the required differential equation .

We know that

→ Here D² - 4D + 4 is a constant ,for remove this constant we have to do double differentiation .

Answer 2

$\tt{\huge{\purple{ ----------- }}}$

QUESTION :

solve

solvethe

solvetheDifferential

solvetheDifferentialequation.

solvetheDifferentialequation.(D² - 4D + 4)y =

solvetheDifferentialequation.(D² - 4D + 4)y =sin2x

ANSWER :

$\to \boxed{\sf{\leadsto{\frac{ {d}^{2} y}{d {x}^{2} } + 4y = 0}}}$

SOLUTION :

Refer to the attachment for the entire solution.