Answer to Example2

Example: Consider a harmonic oscillator for which the differential equation is

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and suppose that mass m=1, the damping constant tex2html_wrap_inline95, and the spring constant tex2html_wrap_inline97. Rewrite this equation as a linear system of differential equations. Solve it, then find the particular solution which satisfies the initial conditions

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Answer. Set v=y'. Then we have

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This gives us the system

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which in matrix form may be rewritten as

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where

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In order to solve this system, we need the characteristic equation

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Its roots are given by the quadratic formulas

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Note that you have to be very careful here since any mistake at finding correctly the roots will generate a far bigger mistakes and waist of time!!
Next we need to find the associated eigenvectors.

Case tex2html_wrap_inline139 . Denote by tex2html_wrap_inline141 the associated eigenvector. The system giving tex2html_wrap_inline143 is

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Since

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(which you should check as an exercise), then the two equations are identical. Hence we take tex2html_wrap_inline149 . If we choose tex2html_wrap_inline151 , we get

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Case tex2html_wrap_inline155 . Similar calculations give the associated eigenvector

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Therefore the general solution is given by

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where tex2html_wrap_inline161 and tex2html_wrap_inline163 are two parameters.

From the above equation giving Y, we may find the solution y to our second differential equation as

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We are almost done except that we need to find the specific solution which satisfies the initial condition

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These two conditions imply

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The second equation gives

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since tex2html_wrap_inline177 , we get tex2html_wrap_inline179 which implies tex2html_wrap_inline181 . Hence we have

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which implies

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and

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which yields

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Next Example:

[Differential Equations] [First Order D.E.]
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Author: Mohamed Amine Khamsi

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