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

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

  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 $\displaystyle \left(x+1\right)\,\left({{d^2}\over{d\,x^2}}\,y\right)+x\,\left({{d }\over{d\,x}}\,y\right)-y=0$ using a first solution $\displaystyle y_{1}=x$ .

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

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

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

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

    $\displaystyle y=e^{x}\,\left(c_{1}\,\sin x+c_{2}\,\cos x\right)$ $\displaystyle y=e^{{{x}\over{2}}}\,\left(c_{1}\,\sin \left({{\sqrt{7}\,x}\over{2 }}\right)+c_{2}\,\cos \left({{\sqrt{7}\,x}\over{2}}\right)\right)$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{5}+1\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(1-\sqrt{5}\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{{{\left(2^{{{3}\over{2}}}+2\right)\,x}\over{2}}}+c_{2} \,e^{{{\left(2-2^{{{3}\over{2}}}\right)\,x}\over{2}}}$

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

    $\displaystyle y=2\,e^{x}\,\sin \left(\sqrt{2}\,x\right)-e^{x}\,\cos \left(\sqrt{2 }\,x\right)$ $\displaystyle y=2\,e^ {- x }\,\sin \left(\sqrt{2}\,x\right)-e^ {- x }\,\cos \left(\sqrt{2}\,x\right)$ $\displaystyle y=2\,e^{x}\,\cos \left(\sqrt{2}\,x\right)-e^{x}\,\sin \left(\sqrt{2 }\,x\right)$ $\displaystyle y=2\,e^ {- x }\,\cos \left(\sqrt{2}\,x\right)-e^ {- x }\,\sin \left(\sqrt{2}\,x\right)$


Department of Mathematics
Last modified: 2026-05-13