Exercise · ZW-UNIT-STEP-FUNCTION-009 · ★★★ Advanced
The solution of \(y'+y=u_1(t)\), \(y(0)=0\), is \(y=u_1(t)\bigl(1-e^{-(t-1)}\bigr)\). What is \(y(2)\)?
Choose one option.
Show answer
- C. \(1-e^{-1}\)
Explanation
At \(t=2\) the step equals 1 and the exponent is \(-(2-1)=-1\). Check: \(Y(s)=\dfrac{e^{-s}}{s(s+1)}\) gives that \(y\).
Review the concept: Unit step function
The unit step function \(u_c(t)\), with \(c\ge0\), equals 0 while \(t<c\) and equals 1 from \(t=c\) on; it is a switch that turns on at time \(c\).
\[u_c(t)=\begin{cases}0,&t
More practice problems: unit step function
- ZW-UNIT-STEP-FUNCTION-001 What is mathcal L\u_3(t)\?
- ZW-UNIT-STEP-FUNCTION-002 What is mathcal L\u_2(t) (t-2)\?
- ZW-UNIT-STEP-FUNCTION-003 If F(s)=mathcal L\f\ and c≥0, what is mathcal L\u_c(t) f(t-c)\?
- ZW-UNIT-STEP-FUNCTION-004 Write, using steps, the function equal to 2 for t<4 and 5 for t≥4.
- ZW-UNIT-STEP-FUNCTION-005 What is mathcal L\u_1(t)-u_3(t)\?
- ZW-UNIT-STEP-FUNCTION-006 What is mathcal L\u_1(t)sin (t-1)\?
- ZW-UNIT-STEP-FUNCTION-007 What is mathcal L^-1\(e^-4s)/(s+1)\?
- ZW-UNIT-STEP-FUNCTION-008 What is mathcal L\t u_2(t)\? (Careful: the factor is t, not t-2.)