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 $\displaystyle y_{1}=x^4$ .
$\displaystyle y=c_{2}\,x^4-{{c_{1}}\over{4}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^7}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^6}}$ $\displaystyle y=c_{1}\,x^4+{{c_{2}}\over{x^5}}$
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 $\displaystyle y_{1}=x+2$ .
$\displaystyle y=c_{2}\,\left(x^2-x+1\right)+c_{1}\,\left(x+2\right)$ $\displaystyle y=c_{2}\,\left(x^2+2\,x+1\right)+c_{1}\,\left(x+2\right)$ $\displaystyle y=c_{2}\,\left(x^2+2\,x-1\right)+c_{1}\,\left(x+2\right)$ $\displaystyle y=c_{2}\,\left(x^2-x-1\right)+c_{1}\,\left(x+2\right)$
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) = $\displaystyle 2-4\,\sqrt{3}$ .
$\displaystyle y=3\,e^{\sqrt{3}\,x-x}-e^{-\sqrt{3}\,x-x}$ $\displaystyle y=3\,e^{x-\sqrt{3}\,x}-e^{\sqrt{3}\,x+x}$ $\displaystyle y=3\,e^{-\sqrt{3}\,x-x}-e^{\sqrt{3}\,x-x}$ $\displaystyle y=3\,e^{\sqrt{3}\,x+x}-e^{x-\sqrt{3}\,x}$
Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y+2\,\left({{d}\over{d\,x}}\,y\right)-3\,y=0$ .
$\displaystyle y=c_{1}\,e^ {- x }+c_{2}\,e^ {- 2\,x }$ $\displaystyle y=c_{1}\,e^{x}+c_{2}\,e^ {- 2\,x }$ $\displaystyle y=c_{1}\,e^ {- x }+c_{2}\,e^ {- 3\,x }$ $\displaystyle y=c_{1}\,e^{x}+c_{2}\,e^ {- 3\,x }$
Solve the second-order linear ODE $\displaystyle {{d^2}\over{d\,x^2}}\,y+2\,\left({{d}\over{d\,x}}\,y\right)+3\,y=0$ .
$\displaystyle y=e^{x}\,\left(c_{1}\,\sin \left(\sqrt{2}\,x\right)+c_{2}\,\cos \left(\sqrt{2}\,x\right)\right)$ $\displaystyle y=\left(c_{2}\,x+c_{1}\right)\,e^{x}$ $\displaystyle y=\left(c_{2}\,x+c_{1}\right)\,e^ {- x }$ $\displaystyle y=e^ {- x }\,\left(c_{1}\,\sin \left(\sqrt{2}\,x\right)+c_{2}\, \cos \left(\sqrt{2}\,x\right)\right)$

Department of Mathematics
Last modified: 2025-08-04