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I was considering the Energy of a driven damped oscillator and came upon the following equation:

given the equation of motion:

[tex]m\ddot x+Dx=-b\dot x+F(t)[/tex]

take the equation multiplied by [tex]\dot x[/tex]

[tex]m\ddot x\dot x+Dx\dot x=-b\dot x^2+F(t)\dot x[/tex]

and we rewrite it:

[tex]\frac{d}{dt}(\frac{m\dot x^2}{2}+\frac{Dx^2}{2})=-b\dot x^2+F(t)\dot x[/tex]

now the LHS appears to be the total energy of the undamped harmonic oscillator, by energy conservation a constant, so it´s rate of change is 0:

[tex]0=-b\dot x^2+F(t)\dot x[/tex]

Ok, so far so good :) But here lies my problem. Now we´ve obtained an equation for \dot x which we could solve for x(t). But the result appears to be different from the standart solution :(

[tex]0=\dot x(-b\dot x+F(t))[/tex]

[tex]\dot x=0[/tex]

[tex]-b\dot x+F(t)=0[/tex]

=> [tex]\dot x_1=const:=C[/tex]

[tex]x(t)=\frac{1}{b}\displaystyle{\int_{t_0}^{t}}F(t')dt'[/tex]

so the solution would read: (would it? Do we have superposition here, since the DE is not linear any more?)

[tex]x(t)=C+\frac{1}{b}\displaystyle{\int_{t_0}^{t}}F(t')dt'[/tex]

So could anyone please help me find the mistake :) I would be thankful :)