Generating...                               quiz07_n20

  1. Find the transformed function Y(s) for the second-order linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y\left(t\right)+y\left(t\right)=e^ {- 2\,t } \,\sinh t$ subject to y(0) = 0 and y'(0) = 1 .

    Y(s) = $\displaystyle {{s^2-3\,s+1}\over{s^4-4\,s^3+4\,s^2-4\,s+3}}$ Y(s) = $\displaystyle {{s^2+4\,s+4}\over{s^4+4\,s^3+4\,s^2+4\,s+3}}$ Y(s) = $\displaystyle {{s^2-4\,s+4}\over{s^4-4\,s^3+4\,s^2-4\,s+3}}$ Y(s) = $\displaystyle {{s^2+5\,s+5}\over{s^4+4\,s^3+4\,s^2+4\,s+3}}$

  2. Find the transformed function Y(s) for the second-order linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y\left(t\right)-2\,\left({{d}\over{d\,t}}\,y \left(t\right)\right)+y\left(t\right)=t$ subject to y(0) = −1 and y'(0) = −1 .

    Y(s) = $\displaystyle -{{s^3-s^2-1}\over{s^4-2\,s^3+s^2}}$ Y(s) = $\displaystyle -{{s^4-s^3-2}\over{s^5-2\,s^4-s^3}}$ Y(s) = $\displaystyle -{{s^3-s^2-1}\over{s^4-2\,s^3-s^2}}$ Y(s) = $\displaystyle -{{s^4-s^3-2}\over{s^5-2\,s^4+s^3}}$

  3. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y\left(t\right)+{{d}\over{d\,t}}\,y\left(t \right)=\sin t$ subject to y(0) = 0 and y'(0) = −1 .

    $\displaystyle y\left(t\right)=-{{\sin t}\over{2}}-{{\cos t}\over{2}}+{{e^ {- t } }\over{2}}$ $\displaystyle y\left(t\right)={{\sin t}\over{2}}-{{\cos t}\over{2}}+{{3\,e^ {- t }}\over{2}}-1$ $\displaystyle y\left(t\right)={{\sin t}\over{2}}-{{\cos t}\over{2}}-{{e^ {- t } }\over{2}}+1$ $\displaystyle y\left(t\right)=-{{\sin t}\over{2}}-{{\cos t}\over{2}}-{{3\,e^ {- t }}\over{2}}+2$

  4. Find the transformed function Y(s) for the second-order linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y\left(t\right)-{{d}\over{d\,t}}\,y\left(t \right)-y\left(t\right)=\cosh t$ subject to y(0) = −1 and y'(0) = 1 .

    Y(s) = $\displaystyle -{{s^3-2\,s^2-2\,s+2}\over{s^4-s^3-2\,s^2+s+1}}$ Y(s) = $\displaystyle -{{s^3-2\,s^2-2\,s+2}\over{s^4-s^3+s-1}}$ Y(s) = $\displaystyle -{{s^3-2\,s^2-2}\over{s^4-s^3+2\,s^2-s+1}}$ Y(s) = $\displaystyle -{{s^3-2\,s^2-2}\over{s^4-s^3-s-1}}$

  5. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y\left(t\right)-{{d}\over{d\,t}}\,y\left(t \right)-y\left(t\right)=e^{t}$ subject to y(0) = −1 and y'(0) = −1 .

    $\displaystyle y\left(t\right)=-e^{t}$ $\displaystyle y\left(t\right)=e^{{{t}\over{2}}}\,\left(-{{\sinh \left({{\sqrt{5... ...2}}\right)}\over{\sqrt{5}}}-\cosh \left({{\sqrt{5}\,t }\over{2}}\right)\right)$ $\displaystyle y\left(t\right)=e^{{{t}\over{2}}}\,\left(-{{2\,\sin \left({{\sqrt... ...t)}\over{\sqrt{3}}}-2\,\cos \left({{\sqrt{3}\,t }\over{2}}\right)\right)+e^{t}$ $\displaystyle y\left(t\right)=e^{{{t}\over{2}}}\,\left(-{{\sin \left({{\sqrt{3}... ...{2}}\right)}\over{\sqrt{3}}}-\cos \left({{\sqrt{3}\,t}\over{2 }}\right)\right)$


Department of Mathematics
Last modified: 2025-05-04