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

    $\displaystyle y=3\,x-x\,\ln x$ $\displaystyle y=3\,x\,\ln x-x$ $\displaystyle y=3\,\cos \ln x-\sin \ln x$ $\displaystyle y=3\,\sin \ln x-\cos \ln x$

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

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

  4. Find a particular solution $y_p$ for the nonhomogeneous linear ODE $\displaystyle {{d^2}\over{d\,t^2}}\,y-{{d}\over{d\,t}}\,y-12\,y=e^{3\,t}$ .

    $\displaystyle {{\left(6\,t-1\right)\,e^{3\,t}}\over{36}}$ $\displaystyle {{\left(2\,t-1\right)\,e^{3\,t}}\over{4}}$ $\displaystyle -{{e^{3\,t}}\over{6}}$ $\displaystyle -{{e^{3\,t}}\over{2}}$

  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: 2025-10-31