Generating...                               quiz08_n16

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

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

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

    $\displaystyle y=2\,x^{\sqrt{3}-1}+3\,x^{-\sqrt{3}-1}$ $\displaystyle y=2\,x^{\sqrt{6}+2}+3\,x^{2-\sqrt{6}}$ $\displaystyle y=3\,x^{\sqrt{3}-1}+2\,x^{-\sqrt{3}-1}$ $\displaystyle y=3\,x^{\sqrt{6}+2}+2\,x^{2-\sqrt{6}}$

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

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

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

    $\displaystyle {{x^3}\over{4}}$ $\displaystyle -{{1}\over{3}}$ $\displaystyle -{{1}\over{2}}$ $\displaystyle {{4\,x^3\,\ln x-x^3}\over{16}}$

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

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


Department of Mathematics
Last modified: 2025-05-04