[Math] First Order Homogeneous Differential Equation

Form

$$ y' = f(y/x) $$

Solving Steps

$$
\begin{split}

y' &= f(y/x) \\
\\
\text{let } u &= y / x \\
y &= u x \\
y' &= u' x + u \\
\\
u' x + u &= f(u) \\
u' x &= f(u) - u \\
\dfrac{du}{dx} x &= f(u) - u \\
\dfrac{1}{f(u) - u} du &= \dfrac{1}{x} dx

\end{split}
$$

Examples

Ex. 1

$$
\begin{split}

\dfrac{dy}{dx} &= \dfrac{ x }{ y} + \dfrac{ y }{ x } \\
&= \dfrac{ x^2 + y^2 }{ x y } \\
&= \cfrac{ \dfrac{1}{x^2} ( x^2 + y^2 ) }{ \dfrac{1}{x^2} ( x y ) } \\
&= \cfrac{ 1 + \dfrac{y^2}{x^2} }{ \dfrac{y}{x} } \\
&= \cfrac{ 1 + \left ( \dfrac{y}{x} \right )^2 }{ \dfrac{y}{x} }

\end{split}
$$

$$
\begin{split}

\text{let } u &= \dfrac{y}{x} \\
y &= u x \\
y' &= u'x + u

\end{split}
$$

$$
\begin{split}

\dfrac{dy}{dx}
&= \cfrac{ 1 + \left ( \dfrac{y}{x} \right )^2 }{ \dfrac{y}{x} } \\
u'x + u &= \dfrac{ 1 + u^2 }{ u } \\
u'x + u &= \dfrac{1}{u} + u \\
u'x &= \dfrac{1}{u} \\
\dfrac{du}{dx} x &= \dfrac{1}{u} \\
u du &= \dfrac{1}{x} dx \\
\int u du &= \int \dfrac{1}{x} dx \\
\dfrac{1}{2} u^2 + C_1 &= \ln \mid x \mid + C_2 \\
u^2 &= 2 \ln \mid x \mid + C \\
u &= \pm \sqrt{ 2 \ln \mid x \mid + C } \\
\dfrac{y}{x} &= \pm \sqrt{ 2 \ln \mid x \mid + C } \\
y &= \pm x \cdot \sqrt{ 2 \ln \mid x \mid + C }

\end{split}
$$

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