Solve the following equation with the initial conditions. x¨ + 4 ˙x + 53x = 15 , x(0) = 8, x˙ = −19

1 answer
Question:

Solve the following equation with the initial conditions. x¨ + 4 ˙x + 53x = 15 , x(0) = 8, x˙ = −19

Answers

[tex]x''+4x'+53x=15[/tex]has characteristic equation[tex]r^2+4r+53=0[/tex]with roots at [tex]r=-2\pm7i[/tex]. Then the characteristic solution is[tex]x_c=C_1e^{(-2+7i)t}+C_2e^{(-2-7i)t}=e^{-2t}\left(C_1\cos(7t)+C_2\sin(7t)\right)[/tex]For the particular solution, consider the ansatz [tex]x_p=a_0[/tex], whose first and second derivatives vanish. Substitute [tex]x_p[/tex] and its derivatives into the equation:[tex]53a_0=15\implies a_0=\dfrac{15}{53}[/tex]Then the general solution to the equation is[tex]x=e^{-2t}\left(C_1\cos(7t)+C_2\sin(7t)\right)+\dfrac{15}{53}[/tex]With [tex]x(0)=8[/tex], we have[tex]8=C_1+\dfrac{15}{53}\implies C_1=\dfrac{409}{53}[/tex]and with [tex]x'(0)=-19[/tex],[tex]-19=-2C_1+7C_2\implies C_2=-\dfrac{27}{53}[/tex]Then the particular solution to the equation is[tex]\boxed{x(t)=\dfrac1{53}e^{-2t}(409cos(7t)-27\sin(7t)+15)}[/tex]

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