First Order ODE/y' = x
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Theorem
The first order ODE:
- $\dfrac {\d y} {\d x} = x$
has the general solution:
- $y = \dfrac {x^2} 2 + C$
Proof
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \rd y\) | \(=\) | \(\ds \int x \rd x\) | Solution to Separable Differential Equation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \dfrac {x^2} 2 + C\) | Primitive of Constant, Primitive of Power |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation