laplacetransform e^t(t+5)^3

解答

拉普拉斯变换

e^{t} (t + 5)^{3}

\frac{125 s ^{3} - 300 s ^{2} + 255 s - 74}{( s - 1 ) ^{4}}

求解步骤

L {e^{t} (t + 5)^{3}}

应用变换法则: if

L {f (t)} = F (s)

则

L {e^{a t} f (t)} = F (s - a)

对于

e^{t} (t + 5)^{3} : f (t) = (t + 5)^{3}, a = 1

= L {(t + 5)^{3}} (s - 1)

L {(t + 5)^{3}} : \frac{6}{s ^{4}} + \frac{30}{s ^{3}} + \frac{75}{s ^{2}} + \frac{125}{s}

= \frac{6}{( s - 1 ) ^{4}} + \frac{30}{( s - 1 ) ^{3}} + \frac{75}{( s - 1 ) ^{2}} + \frac{125}{( s - 1 )}

= \frac{6}{( s - 1 ) ^{4}} + \frac{30}{( s - 1 ) ^{3}} + \frac{75}{( s - 1 ) ^{2}} + \frac{125}{s - 1}

化简

\frac{6}{( s - 1 ) ^{4}} + \frac{30}{( s - 1 ) ^{3}} + \frac{75}{( s - 1 ) ^{2}} + \frac{125}{s - 1} : \frac{125 s ^{3} - 300 s ^{2} + 255 s - 74}{( s - 1 ) ^{4}}

= \frac{125 s ^{3} - 300 s ^{2} + 255 s - 74}{( s - 1 ) ^{4}}

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