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

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

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

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

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

    $\displaystyle {{c_{1}\,\sin \left({{\sqrt{3}\,\ln x}\over{2}}\right)+c_{2}\, \cos \left({{\sqrt{3}\,\ln x}\over{2}}\right)}\over{\sqrt{x}}}$ $\displaystyle {{c_{1}\,\sin \left({{\sqrt{11}\,\ln x}\over{2}}\right)+c_{2}\, \cos \left({{\sqrt{11}\,\ln x}\over{2}}\right)}\over{\sqrt{x}}}$ $\displaystyle x^{{{3}\over{2}}}\,\left(c_{1}\,\sin \left({{\sqrt{3}\,\ln x }\over{2}}\right)+c_{2}\,\cos \left({{\sqrt{3}\,\ln x}\over{2}} \right)\right)$ $\displaystyle c_{1}\,x^{{{\sqrt{5}}\over{2}}+{{3}\over{2}}}+c_{2}\,x^{{{3}\over{2 }}-{{\sqrt{5}}\over{2}}}$

  4. Solve the Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-2\,x\,\left({{d}\over{d\, x}}\,y\right)-2\,y=0$ subject to y(1) = 2 and y'(1) = $\displaystyle 3-2\,\sqrt{17}$ .

    $\displaystyle y=3\,x-{{1}\over{x^2}}$ $\displaystyle y={{3}\over{x^2}}-x$ $\displaystyle y=3\,x^{{{3}\over{2}}-{{\sqrt{17}}\over{2}}}-x^{{{\sqrt{17}}\over{2 }}+{{3}\over{2}}}$ $\displaystyle y=3\,x^{{{\sqrt{17}}\over{2}}+{{3}\over{2}}}-x^{{{3}\over{2}}-{{ \sqrt{17}}\over{2}}}$

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

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

  6. Find a particular solution $y_p$ for the Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-3\,x\,\left({{d}\over{d\, x}}\,y\right)+3\,y=x^5\,e^{x}$ .

    $\displaystyle \left(x^2-2\,x\right)\,e^{x}$ $\displaystyle \left(x^2-x\right)\,e^{x}$ $\displaystyle \left(x^3-3\,x^2+3\,x\right)\,e^{x}$ $\displaystyle x\,e^{x}$


Department of Mathematics
Last modified: 2026-06-26