Question

In the first order differential equation
dy/dx=f (x, y) the function f (x, y) is a function of the ratio y/x:
dy/dx=g (y/x)
Show that the substitution of U=y/x leads to separable equation in U and x.

Answer 1

Answer on Question #63049 – Math – Differential Equations

Question

In the first order differential equation

$$\mathrm{dy/dx} = f(x, y)$$

the function $f(x, y)$ is a function of the ratio $y/x$:

$$\mathrm{dy/dx} = g(y/x)$$

Show that the substitution of $U = y/x$ leads to separable equation in $U$ and $x$.

Solution

$$\frac{dy}{dx} = g\left(\frac{y}{x}\right)$$

Let $\frac{y}{x} = u$, then $y = ux$ and $\frac{dy}{dx} = \frac{d(u \cdot x)}{dx} = x$, and $x \cdot dx = x \cdot \frac{du}{dx} + u$

Replacing $\frac{dy}{dx}$ with $x \cdot \frac{du}{dx} + u$ and $\frac{y}{x} = u$, we get:

$$x \cdot \frac{du}{dx} + u = g(u)$$ $$x \cdot \frac{du}{dx} = g(u) - u$$ $$\frac{du}{g(u) - u} = \frac{dx}{x}.$$

Derived equation is the separable equation in $u$ and $x$.

**Answer**: $\frac{du}{g(u) - u} = \frac{dx}{x}$.

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