Generating...                               quiz04_n28

  1. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y-{{d}\over{d\,x}}\,y-3\,y=0$ .

    $\displaystyle y=\left(c_{2}\,x+c_{1}\right)\,e^ {- x }$ $\displaystyle y=e^{{{x}\over{2}}}\,\left(c_{1}\,\sin \left({{\sqrt{3}\,x}\over{2 }}\right)+c_{2}\,\cos \left({{\sqrt{3}\,x}\over{2}}\right)\right)$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{13}+1\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(1-\sqrt{13}\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{x}+c_{2}\,e^ {- 3\,x }$

  2. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y+3\,\left({{d}\over{d\,x}}\,y\right)-y=0$ subject to y(0) = 2 and y'(0) = $\displaystyle 2\,\sqrt{13}-3$ .

    $\displaystyle y=3\,e^{{{3\,x}\over{2}}-{{\sqrt{13}\,x}\over{2}}}-e^{{{\sqrt{13}\, x}\over{2}}+{{3\,x}\over{2}}}$ $\displaystyle y=3\,e^{{{\sqrt{13}\,x}\over{2}}-{{3\,x}\over{2}}}-e^{-{{\sqrt{13} \,x}\over{2}}-{{3\,x}\over{2}}}$ $\displaystyle y=3\,e^{-{{\sqrt{13}\,x}\over{2}}-{{3\,x}\over{2}}}-e^{{{\sqrt{13} \,x}\over{2}}-{{3\,x}\over{2}}}$ $\displaystyle y=3\,e^{{{\sqrt{13}\,x}\over{2}}+{{3\,x}\over{2}}}-e^{{{3\,x}\over{ 2}}-{{\sqrt{13}\,x}\over{2}}}$

  3. Solve the second-order linear ODE $\displaystyle \left(-x^2-2\,x-3\right)\,\left({{d^2}\over{d\,x^2}}\,y\right)+2\, \left(x+1\right)\,\left({{d}\over{d\,x}}\,y\right)-2\,y=0$ using a first solution $\displaystyle y_{1}=x+1$ .

    $\displaystyle y=c_{2}\,\left(x^2-x-1\right)+c_{1}\,\left(x+1\right)$ $\displaystyle y=c_{2}\,\left(x^2+2\,x-1\right)+c_{1}\,\left(x+1\right)$ $\displaystyle y=c_{2}\,\left(x^2+2\,x-2\right)+c_{1}\,\left(x+1\right)$ $\displaystyle y=c_{2}\,\left(x^2-x-2\right)+c_{1}\,\left(x+1\right)$

  4. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y-9\,y=0$ .

    $\displaystyle y=c_{1}\,e^{2\,x}+c_{2}\,e^ {- 2\,x }$ $\displaystyle y=c_{1}\,e^{3\,x}+c_{2}\,e^ {- 2\,x }$ $\displaystyle y=c_{1}\,e^{3\,x}+c_{2}\,e^ {- 3\,x }$ $\displaystyle y=c_{1}\,e^{2\,x}+c_{2}\,e^ {- 3\,x }$

  5. Solve the second-order linear ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-2\,x\,\left({{d}\over{d\, x}}\,y\right)=0$ using a known solution $\displaystyle y_{1}=x^3$ .

    $\displaystyle y=c_{2}\,x^3-{{c_{1}}\over{3}}$ $\displaystyle y=c_{1}\,x^3+{{c_{2}}\over{x^5}}$ $\displaystyle y=c_{1}\,x^3+c_{2}\,x$ $\displaystyle y=c_{2}\,x^3-{{c_{1}}\over{4\,x}}$


Department of Mathematics
Last modified: 2026-03-24