Linear First Order ODE/dy = f(x) dx/Examples/y' = e^-x^2

Example of Linear First Order ODE: $\d y = \map f x \rd x$

The linear first order ODE:

$(1): \quad \dfrac {\d y} {\d x} = e^{-x^2}$

has the general solution:

$y = \dfrac {\sqrt \pi} 2 \map {\erf} x + C$

where $\erf$ denotes the error function.

Proof

From the definition of the error function:

$\map {\erf} x = \ds \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t$

where $\exp$ is the real exponential function.

So by the Fundamental Theorem of Calculus:

$\dfrac {\rd} {\rd x} \map {\erf} x = \dfrac 2 {\sqrt \pi} e^{-x^2} + C$

The result follows.

$\blacksquare$

Sources

• 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $2$ Fundamental Theorem of the Calculus