First Order ODE/x y' = 2 y

Theorem

The first order ODE:

$x y' = 2 y$

has the general solution:

$y = C x^2$

where $C$ is an arbitrary constant.

Proof

\(\ds x y'\)

\(=\)

\(\ds 2 y\)

\(\ds \leadsto \ \ \)

\(\ds \int \dfrac {\d y} y\)

\(=\)

\(\ds 2 \int \dfrac {\d x} x\)

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds \ln y\)

\(=\)

\(\ds 2 \ln x + \ln C\)

Primitive of Reciprocal

\(\ds \leadsto \ \ \)

\(\ds \ln y\)

\(=\)

\(\ds \map \ln C x^2\)

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds C x^2\)

$\blacksquare$

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: $(3.15)$