1. Find a particular solution $y_p$ for the nonhomogeneous linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y+y=\sin t$ .

    $\displaystyle {{e^ {- t }\,\left(\left(2\,t-1\right)\,e^{2\,t}-2\,t-1\right) }\over{8}}$ $\displaystyle -{{t\,\cos t}\over{2}}$ $\displaystyle {{e^ {- t }\,\left(\left(2\,t-1\right)\,e^{2\,t}+2\,t+1\right) }\over{8}}$ $\displaystyle {{t\,\sin t+\cos t}\over{2}}$

  2. Find a particular solution $y_p$ for the nonhomogeneous linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y-{{d}\over{d\,t}}\,y-6\,y=e^{3\,t}$ .

    $\displaystyle {{e^{3\,t}}\over{5}}$ $\displaystyle \left(-t-1\right)\,e^{3\,t}$ $\displaystyle -e^{3\,t}$ $\displaystyle {{\left(5\,t-1\right)\,e^{3\,t}}\over{25}}$

  3. Solve the Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-2\,x\,\left({{d}\over{d\, x}}\,y\right)+2\,y=0$ subject to y(1) = 5 and y'(1) = 7 .

    $\displaystyle y={{3\,\sin \left({{\sqrt{7}\,\ln x}\over{2}}\right)}\over{\sqrt{x }}}+{{2\,\cos \left({{\sqrt{7}\,\ln x}\over{2}}\right)}\over{\sqrt{ x}}}$ $\displaystyle y=3\,x^2+2\,x$ $\displaystyle y=2\,x^2+3\,x$ $\displaystyle y={{2\,\sin \left({{\sqrt{7}\,\ln x}\over{2}}\right)}\over{\sqrt{x }}}+{{3\,\cos \left({{\sqrt{7}\,\ln x}\over{2}}\right)}\over{\sqrt{ x}}}$

  4. Solve the Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-2\,x\,\left({{d}\over{d\, x}}\,y\right)+y=0$ .

    $\displaystyle x\,\left(c_{2}\,\ln x+c_{1}\right)$ $\displaystyle x^{{{3}\over{2}}}\,\left(c_{1}\,\sin \left({{\sqrt{3}\,\ln x }\over{2}}\right)+c_{2}\,\cos \left({{\sqrt{3}\,\ln x}\over{2}} \right)\right)$ $\displaystyle x\,\left(c_{1}\,\sin \left(\sqrt{2}\,\ln x\right)+c_{2}\,\cos \left(\sqrt{2}\,\ln x\right)\right)$ $\displaystyle c_{1}\,x^{{{\sqrt{5}}\over{2}}+{{3}\over{2}}}+c_{2}\,x^{{{3}\over{2 }}-{{\sqrt{5}}\over{2}}}$

  5. Find a particular solution $y_p$ for the Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-2\,y=x$ .

    $\displaystyle x^2\,\ln x-x^2$ $\displaystyle -{{x}\over{2}}$ $\displaystyle -x\,\ln x-x$ $\displaystyle {{3\,x^2\,\ln x-x^2}\over{9}}$


Department of Mathematics
Last modified: 2026-08-01