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 .

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

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

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

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

    $\displaystyle y\left(t\right)=-{{\sin t}\over{2}}+{{\cos t}\over{2}}-{{e^{t} }\over{2}}$ $\displaystyle y\left(t\right)={{t\,e^{t}}\over{2}}+{{3\,e^{t}}\over{4}}+{{e^ {- t }}\over{4}}-1$ $\displaystyle y\left(t\right)={{t\,e^{t}}\over{2}}-{{5\,e^{t}}\over{4}}+{{e^ {- t }}\over{4}}+1$ $\displaystyle y\left(t\right)=-{{\sin t}\over{2}}+{{\cos t}\over{2}}+{{3\,e^{t} }\over{2}}-2$

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

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

  5. 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) = 0 and y'(0) = 1 .

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


Department of Mathematics
Last modified: 2026-07-07