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 $\displaystyle y_{1}=x^4$ .

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

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

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

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

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

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

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

  5. Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y+{{d}\over{d\,x}}\,y-y=0$ subject to y(0) = 2 and y'(0) = $\displaystyle 2\,\sqrt{5}-1$ .

    $\displaystyle y=3\,e^{-{{\sqrt{5}\,x}\over{2}}-{{x}\over{2}}}-e^{{{\sqrt{5}\,x }\over{2}}-{{x}\over{2}}}$ $\displaystyle y=3\,e^{{{x}\over{2}}-{{\sqrt{5}\,x}\over{2}}}-e^{{{\sqrt{5}\,x }\over{2}}+{{x}\over{2}}}$ $\displaystyle y=3\,e^{{{\sqrt{5}\,x}\over{2}}+{{x}\over{2}}}-e^{{{x}\over{2}}-{{ \sqrt{5}\,x}\over{2}}}$ $\displaystyle y=3\,e^{{{\sqrt{5}\,x}\over{2}}-{{x}\over{2}}}-e^{-{{\sqrt{5}\,x }\over{2}}-{{x}\over{2}}}$


Department of Mathematics
Last modified: 2025-10-15