# Linear First Order ODE/y' = x + y/y(0) = 1

## Theorem

- $(1): \quad \dfrac {\mathrm d y} {\mathrm d x} = x + y$

with initial condition:

- $y \left({0}\right) = 1$

has the particular solution:

- $y = 2 e^x - x - 1$

## Proof

From Linear First Order ODE: $y' = x + y$, the general solution of $(1)$ is:

- $y = C e^x - x - 1$

Setting $y = 1$ when $x = 0$ gives:

- $1 = C + 1$

from which $C = 2$.

Hence the result.

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Appendix $\text{A}$. Numerical Methods