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?