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 $\displaystyle y_{1}=x^4$ .
$\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^5}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^7}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^6}}$ $\displaystyle y=c_{2}\,x^4-{{c_{1}}\over{4}}$
Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y-{{d}\over{d\,x}}\,y+3\,y=0$ subject to y(0) = 2 and y'(0) = $\displaystyle 1-\sqrt{11}$ .
$\displaystyle y=2\,e^{{{x}\over{2}}}\,\cos \left({{\sqrt{11}\,x}\over{2}}\right)- 2\,e^{{{x}\over{2}}}\,\sin \left({{\sqrt{11}\,x}\over{2}}\right)$ $\displaystyle y=2\,e^ {- {{x}\over{2}} }\,\cos \left({{\sqrt{11}\,x}\over{2}} \right)-2\,e^ {- {{x}\over{2}} }\,\sin \left({{\sqrt{11}\,x}\over{2 }}\right)$ $\displaystyle y=2\,e^ {- {{x}\over{2}} }\,\sin \left({{\sqrt{11}\,x}\over{2}} \right)-2\,e^ {- {{x}\over{2}} }\,\cos \left({{\sqrt{11}\,x}\over{2 }}\right)$ $\displaystyle y=2\,e^{{{x}\over{2}}}\,\sin \left({{\sqrt{11}\,x}\over{2}}\right)- 2\,e^{{{x}\over{2}}}\,\cos \left({{\sqrt{11}\,x}\over{2}}\right)$
Solve the second-order linear ODE $\displaystyle \left(-x^2-4\,x-6\right)\,\left({{d^2}\over{d\,x^2}}\,y\right)+2\, \left(x+2\right)\,\left({{d}\over{d\,x}}\,y\right)-2\,y=0$ using a first solution $\displaystyle y_{1}=x+2$ .
$\displaystyle y=c_{2}\,\left(x^2+x-1\right)+c_{1}\,\left(x+2\right)$ $\displaystyle y=c_{2}\,\left(x^2+2\,x-1\right)+c_{1}\,\left(x+2\right)$ $\displaystyle y=c_{2}\,\left(x^2+2\,x-2\right)+c_{1}\,\left(x+2\right)$ $\displaystyle y=c_{2}\,\left(x^2+x-2\right)+c_{1}\,\left(x+2\right)$
Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y-3\,\left({{d}\over{d\,x}}\,y\right)+y=0$ .
$\displaystyle y=c_{1}\,e^{{{\left(\sqrt{5}+3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(3-\sqrt{5}\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{{{\left(2^{{{3}\over{2}}}+2\right)\,x}\over{2}}}+c_{2} \,e^{{{\left(2-2^{{{3}\over{2}}}\right)\,x}\over{2}}}$ $\displaystyle y=c_{1}\,e^{{{\left(\sqrt{13}+3\right)\,x}\over{2}}}+c_{2}\,e^{{{ \left(3-\sqrt{13}\right)\,x}\over{2}}}$ $\displaystyle y=\left(c_{2}\,x+c_{1}\right)\,e^{x}$
Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y-5\,\left({{d}\over{d\,x}}\,y\right)+6\,y=0$ .
$\displaystyle y=c_{1}\,e^{x}+c_{2}\,e^ {- 2\,x }$ $\displaystyle y=c_{1}\,e^{3\,x}+c_{2}\,e^ {- 2\,x }$ $\displaystyle y=c_{1}\,e^{2\,x}+c_{2}\,e^{x}$ $\displaystyle y=c_{1}\,e^{3\,x}+c_{2}\,e^{2\,x}$

Department of Mathematics
Last modified: 2026-10-01