Generating...                               quiz08_n11

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

    $\displaystyle {{1}\over{2}}$ $\displaystyle -{{1}\over{2}}$ $\displaystyle {{\left(3\,x^3\,e^{{{\ln x}\over{2}}}\,\ln x-e^{{{\ln x}\over{2 ... ...n x}\over{2}}}\right)\,\cosh \left({{3\,\ln x}\over{2}}\right)}\over{9\,x^3}}$ $\displaystyle {{1}\over{6\,x}}$

  2. 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) = 2 and y'(1) = $\displaystyle 2-2^{{{3}\over{2}}}$ .

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

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

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

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

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

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

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


Department of Mathematics
Last modified: 2025-09-14