Did you have any careful check again your book or possibly think of something that might be incorrect on the right hand side of your equation anyway ?
If :x: is in place of :t:, it is easily sovable and otherwise, it is going to be different since you have to explain how things are to logically work out in the presence of t.
Okay, assuming you mean dy/dt+ (1/2)y= (1/2)e(t/3), that's a linear equation. Not every equation is "separable". If you are doing first order linear equations you should already have learned about exact equations and "integrating factors".
In this case, the equation is not exact- that is, there is no function f(y) so that d(f(y))/dt= dy/dt+ (1/2)y, but we can find a function u(t) so that, after we multiply the entire equation by u, it is exact. We want to find u(t) so that
d(u(t)y)/dt= u(t)(dy/dt)+ (1/2)u(t)y. Applying the product rule to the left hand side of that, d(u(t)y)/dt= u(t)(dy/dt)+ (du/dt)y= u(t)(dy/dt)+ (1/2)u(t)y.
That means we must have du/dt= (1/2)u which is separable: In fact it is easy to see that u(t)= e(1/2)t.
That tells us that if we multiply the whole equation by e(1/2)t, we get
e(1/2)t(dy/dt)+ (1/2)e(1/2)ty= e(1/2+ (1/3))t
The left hand side is just d(e(1/2)ty)/dt and the right hand side is e(5/6)t. In other words, d(e(1/2)ty)/dt= e(5/6)t.
Integrating both sides, e(1/2)ty= (6/5)e(5/6)t+ C so
y(t)= (6/5)e(1/3)t+ Ce(-1/2)t.
Notice the e(1/3)t in there, exactly as asdf1 suggested.
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