Generating...                               quiz08_n9

  1. Find a particular solution $y_p$ for the nonhomogeneous linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y-2\,\left({{d}\over{d\,t}}\,y\right)-8\,y=e ^{2\,t}$ .

    $\displaystyle -{{e^{2\,t}}\over{8}}$ $\displaystyle {{e^{2\,t}}\over{5}}$ $\displaystyle {{e^{2\,t}}\over{4}}$ $\displaystyle -{{e^{2\,t}}\over{10}}$

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

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

  3. Find a particular solution $y_p$ for the Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-3\,x\,\left({{d}\over{d\, x}}\,y\right)+3\,y=x^4\,e^{x}$ .

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

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

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

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

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


Department of Mathematics
Last modified: 2025-10-30