Consider the equation
We write the characteristic equation
Using the Cardano formulas we get the roots of the equation
"k_2=-1.95-1.31\\times i"
"k_3= -1.95 +1.31\\times i"
where
Therefore solving the equation
"y=C_1\\times e^ {0.9t} + C_2\\times e^ {-1.95t}sin (1.31t)+C_3\\times e^ {-1.95t}cos (1.31t)"
where
are constants. Assume a partial solution of the equation is
where A, B are constants. Hence
"y'''_p=-8Acos(2t)+8Bsin(2t)"
Substituting in the equation we get
As result one gets
"B= \\frac{2} {15}"
Therefore the solution of the equation
is
"y=C_1\\times e^ {0.9t} + C_2\\times e^ {-1.95t}sin (1.31t)+C_3\\times e^ {-1.95t}cos (1.31t)+""- \\frac {1} {15} sin(2t) + \\frac {2} {15} cos(t)"
where
are constants.
Answer.
"y=C_1\\times e^ {0.9t} + C_2\\times e^ {-1.95t}sin (1.31t)+C_3\\times e^ {-1.95t}cos (1.31t)- \\frac {1} {15} sin(2t) + \\frac {2} {15} cos(t)"where
"C_1, C_2, C_3"are constant.
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