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# Examples - Some examples of 2nd order linear ODEs with...

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Unformatted text preview: Some examples of 2nd order linear ODEs with complex and re- peated roots 1 Oscillatory solutions 1.1 Pure oscillations, no damping Let us consider the equation y ′′ + 6 y = 0 , y (0) = 2 , y ′ (0) = 1 The characteristic equation is r 2 + 6 = 0 which has complex conjugate roots r = i √ 6. The roots have zero real part and √ 6 imaginary part and thus solutions are of the form y ( t ) = c 1 cos( √ 6 t ) + c 2 sin( √ 6 t ) We can use the initial conditions to solve for the constants 2 = y (0) = c 1 cos(0)+ c 2 sin(0) = c 1 so c 1 = 2. Also, y ′ ( t ) =- c 1 √ 6 sin( √ 6 t ) + c 2 √ 6 cos( √ 6 t ) so 1 = y ′ (0) =- c 1 √ 6 sin(0) + c 2 √ 6 cos(0) = c 2 √ 6 so c 2 = √ 6 / 6. The solution is thus y ( t ) = 2 cos( √ 6 t ) + √ 6 6 sin( √ 6 t ) plot of the periodic solution 1.2 With damping Now let us see what happens when we add damping. We modify the problem slightly to include damping with coefficient 2, i.e. consider the initial value problem y ′′ + 2...
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Examples - Some examples of 2nd order linear ODEs with...

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