The solution (up to three decimal places) at x=1 of the differential equation {d^{2} y}/{d x^{2}}+2 \frac{d y}{d x}+y=0 subject to boundary conditions y(0)=1 and \frac{d y}{d x}=(0)=-1 is______________.

GATE Civil Engineering (CE) 2018 Shift 1 Solved Paper

Question : 51 of 65

Marks: +1, -0

The solution (up to three decimal places) at

x=1

of the differential equation

\;{d^{2} y}/{d x^{2}}+2 \;{d y}/{d x}+y=0 \;\;

subject to boundary conditions

y(0)=1

and

\;{d y}/{d x}=(0)=-1

is______________.

Solution:

(D^{2}+2 D+1) y=0( ∴

Roots are

-1,-1)

{CF}=(C_{1}+C_{2} x) e^{-x}

y=C_{1} e^{-x}+C_{2} x e^{-x}

.....(i)

y(0)=1

1=C_{1}

....(ii)

y^{'}=C_{1} e^{-x}+C_{2}(e^{-x}-x e^{-x})

y^{'}(0)=-1,-1=C_{1}+C_{2}

....(iii)
From eq. (ii) and (iii),

C_{1}=1, C_{2}=0

∴ \;\; y=e^{-x}

x=1, y=e^{-1}=\;{1}/{e}=0.368

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