Asymptotic behavior of dy/dx = -x - y for large x

[argonurbawono]

is there an exact solution to

dy/dx = -x - y

i am doing a modelling, and just happen to get stumbled into this form of pde.

i do it numerically, but i also want to know how y behave as x approaches large value. i just need to present some analytical work to justify what happen as x grows large.

thanks.

 

[quasar987]

The EDO is linear the exact solution is

$$y(x)=\frac{-\int x e^x dx}{e^x}$$

 

[quasar987]

P.S.

$$\int xe^xdx = e^x(x-1)$$

So y=1-x

 

[HallsofIvy]

Actually, the general solution is y= Ce^-x+ 1- x. You forgot to include the constant of integration.

 

[AlphaNumeric]

Solve the homogeneous equation $$y' + y = 0$$, then consider $$y = Ax+B$$ as the particular solution. When you've got $$\mathcal{L}y = f(x)$$ where f(x) is an n'th order polynomial, trying $$y = a_{n}x^{n} + ... + a_{0}$$ gives the particular solution.

 

[HallsofIvy]

Solve the homogeneous equation $$y' + y = 0$$, then consider $$y = Ax+B$$ as the particular solution. When you've got $$\mathcal{L}y = f(x)$$ where f(x) is an n'th order polynomial, trying $$y = a_{n}x^{n} + ... + a_{0}$$ gives the particular solution.

That's one way to do it. Since this is a first order equation, it's also easy to find an integrating factor, which is what quasar987 did.