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

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

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

    $\displaystyle y=e^ {- {{3\,x}\over{2}} }\,\left(c_{1}\,\sin \left({{\sqrt{3}\,x }\over{2}}\right)+c_{2}\,\cos \left({{\sqrt{3}\,x}\over{2}}\right) \right)$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{5}-3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(-\sqrt{5}-3\right)\,x}\over{2}}}$ $\displaystyle y=e^ {- {{x}\over{2}} }\,\left(c_{1}\,\sin \left({{\sqrt{3}\,x }\over{2}}\right)+c_{2}\,\cos \left({{\sqrt{3}\,x}\over{2}}\right) \right)$ $\displaystyle y=e^ {- {{x}\over{2}} }\,\left(c_{1}\,\sin \left({{\sqrt{11}\,x }\over{2}}\right)+c_{2}\,\cos \left({{\sqrt{11}\,x}\over{2}}\right) \right)$

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

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

  4. Solve the second-order linear ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)+4\,x\,\left({{d}\over{d\, x}}\,y\right)-10\,y=0$ using a known solution $\displaystyle y_{1}=x^2$ .

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

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

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


Department of Mathematics
Last modified: 2024-05-29