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MDoubt No 000017 - GATE


GATE [ME, 2017, 2 marks]
Question : Consider the differential equation $3y''(x)+27y(x)=0$ with initial conditions $y(0)=0$ and $y'(0)=2000$. The value of $y$ at $x=1$ is____.

Solution : Given $3y''(x)+27y(x)=0$
$\implies (3D^2+27)y=0$
The auxiliary equation is $3D^2+27=0$
$\implies D=\pm 3i$
General solution is $y=C_1\cos 3x+C_2\sin 3x\quad\quad\quad\ldots\text{(1)}$
differentiating equation $(1)$
$y'=-3C_1\sin 3x+3C_2\cos 3x$
$y(0)=0$
$\implies C_1=0$
$y'(0)=2000$
$3C_2=2000$
$\implies C_2=\dfrac{200}{3}$
$\therefore$ the particular solution is
$y=\dfrac{2000\sin 3x}{3}$
At $x=1$, $y=\dfrac{2000\sin 3}{3}$
$y=94.08$
CORRECT ANSWER : 94.08
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