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