The solution to the initial value problem is determined as y = e³ˣ.
Form auxiliary equation
The auxiliary equation is determined as follows;
y'' - 7y' + 12y = 0
m² - 7m + 12 = 0
m² - 3m - 4m + 12 = 0
m(m - 3) - 4(m - 3) = 0
m = 3 or 4
[tex]y = c_1e^{3x} + c_2e^{4x}\\\\y(0) = 1\\\\1 = c_1e^{3\times 0} + c_2e^{4\times 0}\\\\1 = c_1 + c_2 \ --(i)\\\\y' = 3c_1e^{3x} + 4c_2e^{4x}\\\\y'(0) = 3\\\\3 = 3c_1e^{3\times 0} + 4c_2e^{4\times 0}\\\\3 = 3c_1 + 4c_2 \ \ --(ii)[/tex]
solve (i) and (ii)
3 = 3c₁ + 4(1 - c₁)
3 = 3c₁ + 4 - 4c₁
3 = -c₁ + 4
c₁ = 1
c₂ = 0
[tex]y = e^{3x}[/tex]
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