1. Solve the second-order linear ODE $\displaystyle x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)+3\,x\,\left({{d}\over{d\, x}}\,y\right)-24\,y=0$ using a known solution $\displaystyle y_{1}=x^4$ .

    $\displaystyle y=c_{2}\,x^4-{{c_{1}}\over{4}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^7}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^5}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^6}}$

  2. 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 $\displaystyle y_{1}=x+2$ .

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

  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) = 2 and y'(0) = $\displaystyle 2-4\,\sqrt{3}$ .

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

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

    $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{13}+3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(3-\sqrt{13}\right)\,x}\over{2}}}$ $\displaystyle y=\left(c_{2}\,x+c_{1}\right)\,e^{x}$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{5}+3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(3-\sqrt{5}\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{{{\left(2^{{{3}\over{2}}}+2\right)\,x}\over{2}}}+c_{2} \,e^{{{\left(2-2^{{{3}\over{2}}}\right)\,x}\over{2}}}$

  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 $\displaystyle y_{1}=x$ .

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


Department of Mathematics
Last modified: 2025-07-04