Generating...                               quiz07_n22

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

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

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

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

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

    $\displaystyle y\left(t\right)={{t\,e^{t}}\over{2}}-{{7\,e^{t}}\over{4}}-{{e^ {- t }}\over{4}}+2$ $\displaystyle y\left(t\right)=-{{\sin t}\over{2}}-{{\cos t}\over{2}}+{{3\,e^{t} }\over{2}}-1$ $\displaystyle y\left(t\right)={{t\,e^{t}}\over{2}}+{{e^{t}}\over{4}}-{{e^ {- t } }\over{4}}$ $\displaystyle y\left(t\right)=-{{\sin t}\over{2}}-{{\cos t}\over{2}}-{{e^{t} }\over{2}}+1$

  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 .

    $\displaystyle y\left(t\right)=e^{{{t}\over{2}}}\,\left(\cosh \left({{\sqrt{5}\,... ...ight)-{{3\,\sinh \left({{\sqrt{5}\,t}\over{2}}\right) }\over{\sqrt{5}}}\right)$ $\displaystyle y\left(t\right)=e^{{{t}\over{2}}}\,\left(2\,\cosh \left({{\sqrt{5... ...{{2\,\sinh \left({{\sqrt{5}\,t}\over{2}}\right) }\over{\sqrt{5}}}\right)-e^{t}$ $\displaystyle y\left(t\right)=e^{t}-{{4\,e^{{{t}\over{2}}}\,\sin \left({{\sqrt{3} \,t}\over{2}}\right)}\over{\sqrt{3}}}$ $\displaystyle y\left(t\right)=e^{{{t}\over{2}}}\,\left(\cos \left({{\sqrt{3}\,t }\over{2}}\right)-\sqrt{3}\,\sin \left({{\sqrt{3}\,t}\over{2}} \right)\right)$


Department of Mathematics
Last modified: 2026-10-28