Linear First Order ODE/y' - (y over x) = 3 x
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Theorem
- $(1): \quad \dfrac {\d y} {\d x} - \dfrac y x = 3 x$
has the general solution:
- $\dfrac y x = 3 x + C$
or:
- $y = 3 x^2 + C x$
Proof
This is an instance of Linear First Order ODE: $y' - \dfrac y x = k x$.
Its solution is:
- $\dfrac y x = k x + C$
or:
- $y = k x^2 + C x$
from which the solution to $(1)$ is found by substituting $3$ for $k$.
$\blacksquare$