Linear First Order ODE/dy = f(x) dx

Theorem

Let $f: \R \to \R$ be an integrable real function.

The linear first order ODE:

$(1): \quad \dfrac {\d y} {\d x} = \map f x$

has the general solution:

$y = \ds \int \map f x \rd x + C$

where $\ds \int \map f x \rd x$ denotes the primitive of $f$.

Initial Condition

Consider the linear first order ODE:

$(1): \quad \dfrac {\d y} {\d x} = \map f x$

subject to the initial condition:

$y = y_0$ when $x = x_0$

$(1)$ has the particular solution:

$y = y_0 + \ds \int_{x_0}^x \map f \xi \rd \xi$

where $\ds \int \map f x \rd x$ denotes the primitive of $f$.

Proof

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \map f x\)

\(\ds \leadsto \ \ \)

\(\ds \int \d y\)

\(=\)

\(\ds \int \map f x \rd x\)

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds \int \map f x \rd x\)

Primitive of Constant

$\blacksquare$

Examples

Example: $y' = e^{-x^2}$

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.

Sources

• 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.1$ Introduction: $(2)$