homogeneous de

Question

To solve a homogeneous DE, we use the substitution

RankGuru · Accepted Answer

1. **Substitution Choice**: For $\frac{dy}{dx} = F(\frac{y}{x})$, we let $y = vx$.
2. **Differentiatiation**: Using the product rule:
   - $\frac{dy}{dx} = v \cdot (1) + x \cdot \frac{dv}{dx} = v + x \frac{dv}{dx}$.
3. **Application**: Substitute both $y$ and $\frac{dy}{dx}$ into the original equation:
   - $v + x \frac{dv}{dx} = F(v)$
   - $x \frac{dv}{dx} = F(v) - v$
4. **Separation**: $\frac{dv}{F(v)-v} = \frac{dx}{x}$, which is now variable separable.

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