Generating...                               quiz04_n15

  1. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y+4\,\left({{d}\over{d\,x}}\,y\right)+3\,y=0$ .

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

  2. Solve the second-order linear ODE $\displaystyle \left(2\,x+1\right)\,\left({{d^2}\over{d\,x^2}}\,y\right)+4\,x\, \left({{d}\over{d\,x}}\,y\right)-4\,y=0$ using a first solution $\displaystyle y_{1}=x$ .

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

  3. Solve the second-order linear ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)-2\,x\,\left({{d}\over{d\, x}}\,y\right)-4\,y=0$ using a known solution $\displaystyle y_{1}=x^4$ .

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

  4. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y-3\,\left({{d}\over{d\,x}}\,y\right)-3\,y=0$ .

    $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{17}+3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(3-\sqrt{17}\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{21}+3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(3-\sqrt{21}\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{17}-3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(-\sqrt{17}-3\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{21}-3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(-\sqrt{21}-3\right)\,x}\over{2}}}$

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

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


Department of Mathematics
Last modified: 2025-06-19