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

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

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

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

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

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

  4. 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 .

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

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

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


Department of Mathematics
Last modified: 2025-08-17