# Differential equation: 1st order

1. Oct 1, 2006

### 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.

2. Oct 1, 2006

### quasar987

The EDO is linear the exact solution is

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

3. Oct 1, 2006

### quasar987

P.S.

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

So y=1-x

4. Oct 4, 2006

### HallsofIvy

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

5. Oct 4, 2006

### 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.

6. Oct 4, 2006

### HallsofIvy

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.