Generating...                               quiz08_n9

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

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

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

    $\displaystyle {{c_{1}\,\ln x}\over{x}}+{{c_{2}}\over{x}}$ $\displaystyle {{c_{1}\,\sin \ln x+c_{2}\,\cos \ln x}\over{x}}$ $\displaystyle x\,\left(c_{1}\,\sin \ln x+c_{2}\,\cos \ln x\right)$ $\displaystyle x\,\left(c_{2}\,\ln x+c_{1}\right)$

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

    $\displaystyle {{x^2}\over{3}}$ $\displaystyle -{{x^2}\over{3}}$ $\displaystyle {{4\,x^3\,\ln x-x^3}\over{16}}$ $\displaystyle {{x^3}\over{8}}$

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

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

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

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


Department of Mathematics
Last modified: 2025-06-19