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Question

# Let a causal LTI system be characterized by the following differential equation, with initial rest condition. d2ydt2+7dydt+10y(t)=4x(t)+5dx(t)dt Where, x(t) and y(t) are the input and output respectively. The impulse response of the system is [u(t) is the unit step function]

A
2e2tu(t)7e5tu(t)
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B
2e2tu(t)+7e5tu(t)
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C
7e2tu(t)2e5tu(t)
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D
7e2tu(t)+2e5tu(t)
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Solution

## The correct option is B −2e−2tu(t)+7e−5tu(t)d2ydt2+7dydt+10y=4x+5dxdt (s2+7s+10)Y(s)=(4+5s)X(s) Y(s)X(s)=5s+4s2+7s+10 Impulse response = L−1(Transfer function) =L−1[5s+4(s+2)(s+5)] =L−1[−2s+2+7s+5] =−2e−2tu(t)+7e5tu(t)

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