Question

find the general solution of differential equation (D^2-4D+2)Y=0​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{(D^2-4\,D+2)y=0}$

$\underline{\textbf{To find:}}$

$\textsf{General solution of the differential equation}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}\;\mathsf{(D^2-4\,D+2)y=0}$

$\textsf{Characteristic equation is}$

$\mathsf{m^2-4m+2=0}$

$\implies\mathsf{m^2-4m+4=2}$

$\implies\mathsf{(m-2)^2=2}$

$\implies\mathsf{m-2=\pm\,\sqrt{2}}$

$\implies\mathsf{m=2\,\pm\,\sqrt{2}}$

$\textsf{Since the roots are real and distinct,}$

$\textsf{the general solution is}$

$\mathsf{y=A\;e^{m_1\,x}+B\;e^{m_2\,x}}$

$\boxed{\mathsf{y=A\;e^{(2+\sqrt{2})x}+B\;e^{(2-\sqrt{2})x}}}$