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

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

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

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

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

    $\displaystyle {{e^ {- 3\,t }}\over{8}}$ $\displaystyle {{e^ {- 3\,t }}\over{14}}$ $\displaystyle {{e^ {- 3\,t }}\over{7}}$ $\displaystyle {{e^ {- 3\,t }}\over{4}}$

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

    $\displaystyle y=-x-{{2}\over{x^2}}$ $\displaystyle y=-2\,x-{{1}\over{x^2}}$ $\displaystyle y=-x^{{{\sqrt{17}}\over{2}}+{{3}\over{2}}}-2\,x^{{{3}\over{2}}-{{ \sqrt{17}}\over{2}}}$ $\displaystyle y=-2\,x^{{{\sqrt{17}}\over{2}}+{{3}\over{2}}}-x^{{{3}\over{2}}-{{ \sqrt{17}}\over{2}}}$

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

    $\displaystyle {{\left(3\,x^3\,e^{{{\ln x}\over{2}}}\,\ln x-e^{{{\ln x}\over{2 ... ...n x}\over{2}}}\right)\,\cosh \left({{3\,\ln x}\over{2}}\right)}\over{9\,x^3}}$ $\displaystyle -{{4\,\ln x+1}\over{16\,x}}$ $\displaystyle {{4\,x^3\,\ln x-x^3}\over{16}}$ $\displaystyle {{x^3}\over{4}}$


Department of Mathematics
Last modified: 2026-03-18