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Question

An input x(t)=e2tu(t) is applied to an LTI system whose impulse response is h(t)=t2etu(t). If the output is in the form y(t)=[Pet+Qtet+Rt2et+Se2t]u(t), then the value of P+Q+R+ S is ________.
  1. -1

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Solution

The correct option is A -1
X(s)=1s+2

H(s)=2(s+1)3

Y(s)=X(s).H(s)=2(s+1)3(s+2)

=As+1+B(s+1)2+C(s+1)3+Ds+2

C=2s+2|s=1=221=2

B=dds[2s+2]|s=1=221=2

A=12!d2ds2[2(s+2)]|s=1=2

Y(s)=2s+12(s+1)2+2(s+1)3+2(s+2)

Y(t)=[2et2tet+t2et2e2t]u(t)

=[Pet+Qtet+Rt2et+Se2t]u(t)

P=2, Q=2, R=1, S=2

P+Q+R+S=22+12=1

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