First Order ODE/dy = k y dx/Proof 1

Theorem

Let $k \in \R$ be a real number.

The first order ODE:

$\dfrac {\d y} {\d x} = k y$

has the general solution:

$y = C e^{k x}$

Proof

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

\(=\)

\(\ds k y\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \int k \rd x\)

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds \ln y\)

\(=\)

\(\ds k x + C'\)

Primitive of Reciprocal, Primitive of Constant

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds e^{k x + C'}\)

\(\ds \)

\(=\)

\(\ds e^{k x} e^{C'}\)

\(\ds \)

\(=\)

\(\ds C e^{k x}\)

putting $C = e^{C'}$

$\blacksquare$

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