First Order ODE/y' = e^x
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Theorem
The first order ODE:
- $y' = e^x$
has the general solution:
- $y = e^x + C$
where $C$ is an arbitrary constant.
Proof
\(\ds y'\) | \(=\) | \(\ds e^x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \d y\) | \(=\) | \(\ds \int e^x \rd x\) | Solution to Separable Differential Equation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds e^x + C\) | Primitive of Constant, Primitive of Exponential Function |
$\blacksquare$
Sources
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: $(3.12)$