Generating...                               quiz08_n21

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

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

  2. Find a particular solution $y_p$ for the Cauchy-Euler ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-3\,x\,\left({{d}\over{d\, x}}\,y\right)+3\,y=x^5\,e^{x}$ .

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

  3. Find a particular solution $y_p$ for the nonhomogeneous linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y-5\,\left({{d}\over{d\,t}}\,y\right)+6\,y=e ^{3\,t}$ .

    $\displaystyle {{e^{3\,t}}\over{6}}$ $\displaystyle \left(-t-1\right)\,e^{3\,t}$ $\displaystyle \left(t-1\right)\,e^{3\,t}$ $\displaystyle -{{e^{3\,t}}\over{6}}$

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

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

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

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


Department of Mathematics
Last modified: 2026-07-16