Generating...                               quiz04_n23

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

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

  2. Solve the second-order linear ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)+4\,x\,\left({{d}\over{d\, x}}\,y\right)-18\,y=0$ using a known solution $\displaystyle y_{1}=x^3$ .

    $\displaystyle y=c_{1}\,x^3+{{c_{2}}\over{x^5}}$ $\displaystyle y=c_{1}\,x^3+{{c_{2}}\over{x^6}}$ $\displaystyle y=c_{1}\,x^3+{{c_{2}}\over{x^3}}$ $\displaystyle y=c_{2}\,x^3-{{c_{1}}\over{3}}$

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

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

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

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

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

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


Department of Mathematics
Last modified: 2026-05-20