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Explain why differentiation of y with respect to x is done on the left side but on the right differentiation with respect to x is done?

In the method shown, the equation involves two variables \( x \) and \( y \), and the functions \( f_1(x, y) \) and \( f_2(x, y) \). These are homogeneous functions of the same degree. To solve the differential equation, the goal is to reduce the equation into a form where it can be solved through integration.

Here's why differentiation is done on the left side with respect to \( x \):

1. **Substitution \( y = vx \):**
   To simplify the equation, the substitution \( y = vx \) is made, where \( v \) is a new variable that depends on \( x \). This substitution is based on the homogeneity of the functions \( f_1(x, y) \) and \( f_2(x, y) \). Since \( y \) is expressed in terms of \( v \) and \( x \), you need to differentiate \( y = vx \) with respect to \( x \) to find the derivative \( \frac{dy}{dx} \).

2. **Differentiating \( y = vx \):**
   Applying the product rule of differentiation:
   \[
   \frac{dy}{dx} = v + x \frac{dv}{dx}
   \]
   This gives the relationship between \( \frac{dy}{dx} \), \( v \), and \( \frac{dv}{dx} \). It allows us to replace \( \frac{dy}{dx} \) in the original equation with terms involving \( v \) and \( \frac{dv}{dx} \), making it easier to separate variables.

3. **Substitution into the original equation:**
   After differentiating \( y = vx \), the result \( \frac{dy}{dx} = v + x \frac{dv}{dx} \) is substituted back into the original differential equation. Since both \( f_1(x, y) \) and \( f_2(x, y) \) are functions of \( x \) and \( y \), we replace \( y \) with \( vx \), transforming the equation into terms of \( x \) and \( v \).

4. **Separation of variables:**
   The goal of this substitution is to reduce the equation to a form where the variables can be separated, allowing us to integrate both sides. The final equation becomes:
   \[
   \frac{dv}{dx} = \frac{f_1(x, vx)}{f_2(x, vx)} - v
   \]
   Now \( \frac{dv}{dx} \) is written as a function of \( x \) and \( v \), and we can proceed by integrating both sides to find the solution.

### Key Point:
Differentiating \( y = vx \) allows you to express \( \frac{dy}{dx} \) in terms of \( v \) and \( x \), transforming the equation into a solvable form. The original equation involves \( y \), but after substitution, it is rewritten entirely in terms of \( v \) and \( x \), allowing you to integrate and solve the differential equation.

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### Related Questions:
1. What are homogeneous functions and how do they help in solving differential equations?
2. Can you explain the steps to separate variables after substitution?
3. What is the significance of integrating both sides of the equation in this method?
4. How is the integration constant \( C \) found in such solutions?
5. How does the degree of homogeneity affect the solution process?

### Tip:
In homogeneous differential equations, converting to new variables using substitutions like \( y = vx \) simplifies the equation by exploiting the properties of the functions involved, making them easier to solve.

Explore the differentiation steps in solving homogeneous differential equations. Learn why y = vx substitution simplifies the equation and how differentiation with respect to x is applied in this method. Suitable for Grades 11-12 students.

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