Differential Equations

Differential Equations - Differential Equations Important...

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Differential Equations –Important Applications First Order Unforced: τ dx/dt + x = 0 Solution: x = x(0)*exp( -t/ τ ) where x(0)= initial condition at t=0 Forced: τ dx/dt + x = x in x in = constant input forcing Solution: x=x(0) *exp(-t/ τ ) + x in *(1-exp(-t/ τ )) where x(0)=initial condition at t=0, if x(0)=0 then the first term vanishes. Numerical solution form: dx/dt = (x in -x)/ τ x(0)=initial condition Second Order Unforced: d 2 x/dt 2 + 2* ζϖ n *dx/dt+ ϖ n 2 *x = 0 for dx/dt(0)=0 Solution: x=x(0)*exp(- ζϖ n *t)*(cos( ϖ r *t)+ ζ *sin( ϖ r *t)/sqrt(1- ζ 2 )) where ϖ r = ϖ n*sqrt(1- ζ 2 ) Forced: d 2 x/dt 2 + 2* ζϖ n *dx/dt+ ϖ n 2 *x = x in * ϖ n 2 x in =constant input forcing Solution: x=unforced + x in *(1- exp(-

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Differential Equations - Differential Equations Important...

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