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Ethiopia
Part II: Fill the Blank Space with the 4. Let \( f(t)=\left\{\begin{array}{rr}0, & 0 \leq t<3 \\ 2, & t \geq 3\end{array}\right. \), then \( L \) 5. \( L^{-1}\left\{\frac{-2 s+6}{s^{2}+4}\right\}= \) 6. \( L\left\{t^{n} f(t)\right\}= \) 7. The particular solution of \( y^{\prime \prime}+2 y= \) 8. The indicated \( y_{1}(x)=x e^{-x} \) is a solu \( y^{\prime \prime}+2 y^{\prime}+y=0 \), then \( y_{2}(x)= \)
Calculus Jan 20, 2025
WEIGHT: \( 50 \% \) January 20/1/2025 Name FIKADU BURKA Part I: Write TRUE or FALSE on th 1. The general solution of \( y^{\prime \prime}-y=0 \) 2. The equation \( y^{\prime \prime}-y=e^{x} \) has a pa \( y_{p}=A e^{x} \) 3. \( L\left\{t^{4} e^{7 t}\right\}=\frac{\sqrt{5}}{(s-7)^{5}} \)
Calculus Jan 20, 2025
WEIGHT: 50\% SEMESTER: 2 January 20/1/2025 Time allowed 3:00 hour Name \( \qquad \) FIKADU Burka Id. Number ZD 1403666 Part I: Write TRUE or FALSE on the provided space only(6pts). 1. The general solution of \( y^{\prime \prime}-y=0 \) can be written as \( y=c_{1} \cosh x+c_{2} \sinh x \) \( \qquad \) 2. The equation \( y^{\prime \prime}-y=e^{x} \) has a particular solution of the form \[ y_{p}=A e^{x} \] \( \qquad \) 3. \( L\left\{t^{4} e^{7 t}\right\}=\frac{\sqrt{5}}{(s-7)^{5}} \) \( \qquad \) Part II: Fill the Blank Space with the short \& simplified answer. (20pts) 4. Let \( f(t)=\left\{\begin{array}{lr}0, & 0 \leq t<3 \\ 2, & t \geq 3\end{array}\right. \), then \( L\{f(t)\}= \) \( \qquad \) 5. \( L^{-1}\left\{\frac{-2 s+6}{s^{2}+4}\right\}= \) \( \qquad \) 6. \( L\left\{t^{n} f(t)\right\}= \) \( \qquad \) 7. The particular solution of \( y^{\prime \prime}+2 y=-4 x+10 \) is \( \qquad \) 8. The indicated \( y_{1}(x)=x e^{-x} \) is a solution of the given differential equation, \( y^{\prime \prime}+2 y^{\prime}+y=0 \), then \( y_{2}(x)= \) \( \qquad \) 1 Page
Calculus Jan 20, 2025
Part III: Show all the necessary steps neatly and clearly. (24pts) 10. Show that \( L\left\{f^{\prime \prime \prime}(t)\right\}=s^{3} F(s)-s^{2} f(0)-s f^{\prime}(0)-f^{\prime \prime}(0) \)
Calculus Jan 20, 2025
7. If \( f(x)=\log 2^{x^{2}+4 x-5} \) then, what is the domain of \( \mathrm{f}(\mathrm{x}) \) ?
Pre Algebra Jan 20, 2025
\( f(x)=5x^{2}+2x+1=0 \)
Algebra Jan 19, 2025
\( 25^{1-x=\sqrt {125}} \)
Algebra Jan 19, 2025
The Initial Value Froblem by Laplace TVansfornes a) \( y^{\prime}+3 y=t, y(0)=2 \) b) \( y^{\prime \prime}+2 y^{\prime}+5 y=e^{-t} \sin t, y(0)=0, y^{\prime}(0)=1 \) c) \( y^{\prime \prime}-6 y^{\prime}+9 y=t^{2} e^{3 t}, y^{\prime}(0)=6, y(0)=2 \)
Calculus Jan 19, 2025
The Initial Value Froblem by Laplace. TVansformes a) \( y^{\prime}+3 y=t, y(0)=2 \) b) \( y^{\prime \prime}+2 y^{\prime}+5 y=e^{-t} \sin t, y(0)=0, y^{\prime}(0)=1 \) c) \( y^{\prime \prime}-6 y^{\prime}+9 y=t^{2} e^{3 t}, y^{\prime}(0)=6, y(0)=2 \)
Calculus Jan 19, 2025
The Initial Value Froblem by Laplace. TVanBformes a) \( y^{\prime}+3 y=t, y(0)=2 \) b) \( y^{\prime \prime}+2 y^{\prime}+5 y=e^{-t} \sin t, y(0)=0, y^{\prime}(0)=1 \) c) \( y^{\prime \prime}-6 y^{\prime}+9 y=t^{2} e^{3 t}, y^{\prime}(0)=6, y(0)=2 \) Lif'(t)=sLSt(t)-th) LIf \( (t)=L L+2 \)
Calculus Jan 19, 2025
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