2nd Order Vector Differential Equation

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φM23
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If I have a particular integral r = Ce2t and dr/dt = 2Ce2t , why do I have to differentiate this again and substitute into the given differential equation? Why can't I use the initial conditions dr/dt = (i - j) and t=0 to find C using the first derivative of the particular integral?

Thanks for any help!
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Farhan.Hanif93
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(Original post by PhyM23)
If I have a particular integral r = Ce2t and dr/dt = 2Ce2t , why do I have to differentiate this again and substitute into the given differential equation? Why can't I use the initial conditions dr/dt = (i - j) and t=0 to find C using the first derivative of the particular integral?

Thanks for any help!
That's because the initial conditions apply to the general solution \mathbf{r}(t)=\mathbf{r}_c(t)+ \mathbf{r}_p(t) (where subscripts c and p denote the complementary and particular solutions respectively), whilst the particular integral is not the general solution.

Furthermore, the purpose of the particular integral is to give a solution of the differential equation, and without direct substitution (i.e. by computing the second derivative in your case) this is unverified. In general, the constants in the particular integral are to be determined from the differential equation itself, whilst the constants in the complementary function are to be determined by the initial conditions for the general solution [in that order].*
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φM23
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(Original post by Farhan.Hanif93)
That's because the initial conditions apply to the general solution \mathbf{r}(t)=\mathbf{r}_c(t)+ \mathbf{r}_p(t) (where subscripts c and p denote the complementary and particular solutions respectively), whilst the particular integral is not the general solution.

Furthermore, the purpose of the particular integral is to give a solution of the differential equation, and without direct substitution (i.e. by computing the second derivative in your case) this is unverified. In general, the constants in the particular integral are to be determined from the differential equation itself, whilst the constants in the complementary function are to be determined by the initial conditions for the general solution [in that order].*
Just what I needed. Thank you for your help
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