Learning DEs: Solving 2nd Order Differential Equations

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To solve the second-order differential equation derived from Newton's second law, x'' = F/m, one can integrate the equation twice to obtain the equation of motion. The first integration gives the velocity function, and the second integration leads to the position function, x(t). This approach simplifies the process compared to more complex methods often found in textbooks. The discussion highlights that the initial confusion stemmed from the focus on homogeneous equations rather than applying basic integration techniques. Ultimately, integrating twice provides a straightforward solution to the problem.
greg_rack
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Hi guys,

I have just started studying DEs on my own, so pardonne moi in advance for the probably silly question :)

Via Newton's second law of motion:
$$x''=\frac{F}{m} \ [1]$$
Which is a second-order differential equation.
But, from here, how do I get the good old equation of motion:
$$x(t)=\frac{F}{2m}t^2+vt+x$$
by solving the DE? What is the procedure to apply? In my textbook, only second-order homogeneous DE are treated, but nothing with the form of ##[1]##... and online everything looks over-complicated.
 
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Ok, I managed to get to the solution just by integrating twice both sides.
I was wrapping my head for nothing!
 

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