Question

the solution of differential equation d³y/dx³ -d²y/dx²+4 dy/dx - 4y =0 is​

Answer 1

SOLUTION

TO DETERMINE

The solution of

$\displaystyle \sf{ \frac{ {d}^{3} y}{d {x}^{3} } - \frac{ {d}^{2} y}{d {x}^{2} } + 4 \frac{ {d}^{} y}{d {x}^{} } - 4y = 0}$

EVALUATION

Here the given differential equation is

$\displaystyle \sf{ \frac{ {d}^{3} y}{d {x}^{3} } - \frac{ {d}^{2} y}{d {x}^{2} } + 4 \frac{ {d}^{} y}{d {x}^{} } - 4y = 0}$

$\sf{Let \: \: y = {e}^{mx} \: \: be \: the \: trial \: solution }$

Then the auxiliary equation is

$\sf{ {m}^{3} - {m}^{2} + 4m - 4= 0}$

$\sf{ \implies \: (m - 1)( {m}^{2} + 4)= 0}$

$\sf{ \implies \: m = 1 \: , \: 2i \: ,\: - 2i}$

Hence the required solution is

$= \sf{a {e}^{x} + b \cos 2x + c \sin 2x }$

Where a , b , c are constants

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