First Order ODE/y' = x

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