Generating...                               quiz07_n26

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

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

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

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

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

    $\displaystyle y\left(t\right)={{e^{2\,t}}\over{3}}-{{e^ {- t }}\over{3}}$ $\displaystyle y\left(t\right)={{2\,e^{{{t}\over{2}}}\,\sin \left({{\sqrt{7}\,t }\over{2}}\right)}\over{\sqrt{7}}}$ $\displaystyle y\left(t\right)=-{{2\,e^{{{t}\over{2}}}\,\sin \left({{\sqrt{7}\,t }\over{2}}\right)}\over{\sqrt{7}}}$ $\displaystyle y\left(t\right)={{e^ {- t }}\over{3}}-{{e^{2\,t}}\over{3}}$

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

    Y(s) = $\displaystyle -{{s^2+3\,s+3}\over{s^4+4\,s^3+6\,s^2+4\,s+5}}$ Y(s) = $\displaystyle -{{s^2-4\,s+2}\over{s^4-4\,s^3+4\,s^2-4\,s+3}}$ Y(s) = $\displaystyle -{{s^2+4\,s+2}\over{s^4+4\,s^3+4\,s^2+4\,s+3}}$ Y(s) = $\displaystyle -{{s^2-5\,s+7}\over{s^4-4\,s^3+6\,s^2-4\,s+5}}$

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

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


Department of Mathematics
Last modified: 2025-06-19