Is There an Exact Solution for the Differential Equation y'= 2 + (1+sin(t))y/5?

[kochibacha]

it's linear ordinary differential equation and when i tried to solve the integrating factor cannot be put in an elementary form. The text i that i got this equation tell me to plot y versus t for several constant C so there must be the exact solution

 

[Simon Bridge]

... when i tried to solve the integrating factor cannot be put in an elementary form.

Sure it can: http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

y'=2+(1+sint)y

can be rearranged so it looks like:

y'+p(t)y=g(t)

 

[HallsofIvy]

The point of an integrating factor, \alpha, is that it allows us the write the parts including y as a single derivative: (\alpha y)'= \alpha y'+ \alpha' y and here we want that to equal y'- (1+ sin(t))y. That is, we want \alpha'= d\alpha/dt= -1 - sin(t) which is "separable".

 

[HallsofIvy]

The point of an integrating factor, \alpha, is that it allows us the write the parts including y as a single derivative: (\alpha y)'= \alpha y'+ \alpha' y and here we want that to equal y'- (1+ sin(t))y. That is, we want \alpha'= d\alpha/dt= -1 - sin(t) which is "separable".

I should have said
\frac{d\alpha}{dt}= (-1- sin(t))\alpha

 

[Simon Bridge]

@kochibacha: any of this useful?