Generating...                               quiz07_n29

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

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

  2. 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 .

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

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

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

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

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

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

    Y(s) = $\displaystyle -{{s^3-4\,s^2+4\,s+2}\over{s^4-5\,s^3+7\,s^2+3\,s-10}}$ Y(s) = $\displaystyle -{{s^3+4\,s^2+4\,s-2}\over{s^4+3\,s^3-s^2-13\,s-10}}$ Y(s) = $\displaystyle -{{s^3-4\,s^2+3\,s-1}\over{s^4-5\,s^3+5\,s^2+5\,s-6}}$ Y(s) = $\displaystyle -{{s^3+4\,s^2+3\,s-1}\over{s^4+3\,s^3-3\,s^2-11\,s-6}}$


Department of Mathematics
Last modified: 2026-05-20