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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# General solution of double integrals

**Physics Forums | Science Articles, Homework Help, Discussion**