GATE Mechanical Engineering (ME) 2018 Shift 2 Solved Paper

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Question : 46 of 65

Marks: +1, -0

Given the ordinary differential equation

{d^{2} y}/{d x^{2}}+\;{d y}/{d x}-6 y=0

with

y(0)=0

and

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

, the value of

y(1)

is (correct to two decimal places).

Your Answer:

Solution:

(D^{2}+D-6) y=0

y(0)=0

y(0)=1

(D+3)(D-2) y=0

D=2,-3

C.

F_{0}=C_{1} e^{2 x}+C_{2} e^{-3 x}

y=C_{1} ∈^{2 x}+C_{2} e^{-3 x}

y(0)=0

So,

0=C_{1}+C_{2}

...(i)

\;{d y}/{d x}=2 C_{1} e^{2 x}-3 C_{2} e^{-3 x}

y{'}(0)=1

1=2 C_{1}-3 C_{2}

....(ii)
From equation (i) and (ii),

C_{1}=\;{1}/{5}

C_{2}=\;{-1}/{5}

y=\;{1}/{5} e^{2 x}-\;{1}/{5} e^{-3 x}

When,

x=1

y(1)=\;{e^{2}-e^{-3}}/{5}=1.4678

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