I have a differential equation of the form and I want to solve it using calculus, as opposed to using a differential equation method.(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\frac{d^2v}{dt}[/itex] = [itex]\alpha[/itex]

where [itex]v[/itex] is a function of [itex]t[/itex] i.e., [itex]v(t)[/itex]

and [itex]\alpha[/itex] is some constant.

How do I solve for [itex]v(t)[/itex] if the time ranges from [itex]t_0[/itex] to [itex]t [/itex] such that [itex]v'(t_0)[/itex] and [itex]v(t_0)[/itex] are the lower bounds of my double integrals?

Do I setup a double integral, such as the following?

[itex]\int_{v(t_0)}^{v(t)}[/itex] [itex]\int_{v'(t_0)}^{v(t)} dvdv [/itex] = [itex]\alpha[/itex][itex]\int_{t_0}^{t}[/itex] [itex]\int_{t_0}^{t} dτ dτ [/itex]

I get the following solution from evaluating that double integral:

[itex]v(t) = \frac{\alpha}{2}{(t-t_0})^2 +v'(t_0)(t-t_0) + v(t_0)[/itex]

and I was wondering if the solution is correct?

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# General solution of double integrals

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