Convert 2nd order differential equation to 1st order

Question

I don't understand how to convert the following 2nd order equation to a first order equation:

x''(t) = 686 -0.5*x'(t ^2) + 10(x(t) + 150)

I used the substitution method but am not sure how to convert x(t):

x'(t) = y(t),

x''(t) = y'(t),

x(t) = y^2(t)/2,

y'(t) = 686 -0.5*y(t ^2) + 10(y^2(t)/2 + 150)

Answer 1

This transformation is purely mechanical. You inserted an unjustified step in x(t)=y^2(t)/2 that is unconnected to the existing equations. You rightly set x'(t)=y(t), and after that it is just inserting

y'(t) = x''(t) = 686 -0.5*y(t)^2 + 10*(x(t) + 150)