Question

Section 5.2 Problem 7:Find the general solution3D%200" id="TexFormula1" title="y'' - 8y' + 16y = 0" alt="y'' - 8y' + 16y = 0" align="absmiddle" class="latex-formula">
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Answer 1

Answer:

$y(x)=C_1e^{4x}+C_2e^{4x}$

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:

$y''-8y'+16y=0\\\\m^2-8m+16=0\\\\(m-4)^2=0\\\\m-4=0\\\\m=4$

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

Thus, the general solution for our given differential equation is $y(x)=C_1e^{4x}+C_2e^{4x}$