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Prove a method to obtain a first-order, separable differential equation

Prove that the substitution y = \frac{x}{v} transforms a homogeneous equation y' = f(x,y) into a first-order, separable equation for v.


Proof. Let

    \[ y = \frac{x}{v} \quad \implies \quad y' = \frac{v - xv'}{v^2}. \]

Since

    \[ f(x,y) = f \left(x, \frac{x}{v} \right) \]

is homogeneous, we have

    \[ f \left( xt, \frac{x}{v} t \right) = f\left( x, \frac{x}{v} \right). \]

Letting t = \frac{1}{x} we then have

    \[ f(x,y) = f \left( 1, \frac{1}{v} \right). \]

So,

    \begin{align*}  y' = f(x,y) && \implies && \frac{v-xv'}{v^2} &= f \left( 1, \frac{1}{v} \right) \\  && \implies && v'x &= v - v^2 f \left( 1, \frac{1}{v} \right) \\  && \implies && \frac{1}{v - v^2 f \left( 1, \frac{1}{v} \right)} v' &= \frac{1}{x}.  \end{align*}

Hence, this equation is separable. \qquad \blacksquare

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