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boneill3
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Homework Statement
Solve [itex]y''+4y'+5y=0[/itex]
find solutions for y(0)=1 and y'(0)=0
Homework Equations
Quadratic equation
The Attempt at a Solution
Hows this look ?
assume solution is in the form of [itex]y=ce^{rx}[/itex]
substitute [itex]y=ce^{rx}[/itex] into the equation.
[itex]cr^2e^{rx}+4cre^{rx}+5ce^{rx}=0[/itex]
we then divide by [itex]ce^{rx}[/itex] to give
[itex]r^2+4r+5=0[/itex]
So we than find the roots which are [itex]r_1=-2+i,r_2=-2-i[/itex]
which gives the solution of
[itex]y=c_1e^{-2+ix}+c_2e^{-2-ix}[/itex].
but as they are complex roots we use the equation
[itex]y=c_1e^{\lambda x}cos \mu x +c_2e^{\lambda x}sin \mu x }[/itex].
where [itex]\mu = 1 , \lambda = -2[/itex]
so
[itex]y(x)=c_1e^{-2x}cos x +c_2e^{-2x}sin x [/itex].
and
[itex]y'(x)=e^{-2x}(sin (x)(-c_1-sc_2)+cos(x)(c_2-2c_1)) [/itex].
to find [itex]c_1[/itex] and [itex]c_2[/itex] apply the initial conditions.
[itex]y(0) = c_1 = 1[/itex]
and
[itex]y'(0) = c_2-2= 0 [/itex] so [itex]c_2 = 2[/itex]
so that makes the general solution
[itex]y(x)=e^{-2x}cos x +2e^{-2x}sin x [/itex].
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