Answer:
[tex]y(x)=C_1e^{-x}+C_2e^{3x}[/tex]
Step-by-step explanation:
To solve a second-order homogeneous differential equation, we need to substitute each term with the auxiliary equation [tex]am^2+bm+c=0[/tex] where the values of [tex]m[/tex] are the roots:
[tex]y''-2y'+3y=0\\\\m^2-2m+3=0\\\\(m+1)(m-3)=0\\\\m=-1,\:m=3[/tex]
Since the values of [tex]m[/tex] are distinct real roots, then the general solution is [tex]y(x)=C_1e^{m_1x}+C_2e^{m_2x}[/tex].
Thus, the general solution for our given differential equation is [tex]y(x)=C_1e^{-x}+C_2e^{3x}[/tex].
