Question

Section 5.2 Problem 9:Find the general solution.3D%200" id="TexFormula1" title="y'' - 2y' + 3y = 0" alt="y'' - 2y' + 3y = 0" align="absmiddle" class="latex-formula">
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Answer 1

Answer:

$y(x)=C_1e^{-x}+C_2e^{3x}$

Step-by-step explanation:

To solve a second-order homogeneous differential equation, we need to substitute each term with the auxiliary equation $am^2+bm+c=0$ where the values of $m$ are the roots:

$y''-2y'+3y=0\\\\m^2-2m+3=0\\\\(m+1)(m-3)=0\\\\m=-1,\:m=3$

Since the values of $m$ are distinct real roots, then the general solution is $y(x)=C_1e^{m_1x}+C_2e^{m_2x}$.

Thus, the general solution for our given differential equation is $y(x)=C_1e^{-x}+C_2e^{3x}$.