Generating...                               quiz08_n0

  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 Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-2\,y=1$ .

    $\displaystyle -{{x}\over{4}}$ $\displaystyle -{{1}\over{2}}$ $\displaystyle -{{1}\over{3}}$ $\displaystyle -{{x}\over{2}}$

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

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

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

    $\displaystyle {{e^ {- 3\,t }}\over{28}}$ $\displaystyle -{{\left(7\,t+1\right)\,e^ {- 3\,t }}\over{49}}$ $\displaystyle {{e^ {- 3\,t }}\over{8}}$ $\displaystyle -{{\left(2\,t+1\right)\,e^ {- 3\,t }}\over{4}}$

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

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


Department of Mathematics
Last modified: 2026-07-16