dy/dt = -t^3y^2-ty^2
dx/dt=(sin t)/(cos t+10)
dy/dx -3y/x =x^3cosx
Which differential method should be used for each equation?
I'd say
separation of variables, since you can write it as
dy/y^2 = -(t+t^3) dt
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next is just direct integration, since you already have dy/dt as a function of t.
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The last is of the form
y' + y p(x) = q(x)
so look for an integrating factor