Solve y^2zp + z^2xq = y^2x
Understand the Problem
The question involves solving a mathematical equation related to the method of grouping and multipliers in differential equations. It appears to seek a solution to a specific equation in the context of these methods.
Answer
The general solution is given by the equation: $y^3 - 3zx^2 = C$.
Answer for screen readers
The general solution is given by the equation: $$ y^3 - 3zx^2 = C $$
Steps to Solve
- Identify the equation form
The given equation is of the form $y^2 z p + z^2 x q = \bar{y} y^2 x$, where $p = y^2$, $q = 2zx$, and $r = y^2 x$.
- Rewrite the equation
We can rearrange the equation to understand the relationships between the terms: $$ \frac{dx}{y^2} = \frac{dy}{2zx} = \frac{dz}{y^2x} $$
- Separate the variables
Now, we can separate the variables in the given equation. For instance, from the first two fractions: $$ \frac{dx}{y^2} = \frac{dy}{2zx} $$ Cross multiply: $$ 2zx , dx = y^2 , dy $$
- Integrate both sides
Now, integrate both sides: $$ \int 2zx , dx = \int y^2 , dy $$
- Perform the integration
Carrying out the integration we get: $$ zx^2 = \frac{y^3}{3} + C $$
- Re-express the solution
Rearranging this gives: $$ y^3 = 3zx^2 + C $$
- Final solution format
Thus, the general solution can be expressed as: $$ y^3 - 3zx^2 = C $$
The general solution is given by the equation: $$ y^3 - 3zx^2 = C $$
More Information
This type of differential equation shows how variables can interact with each other. The approach here involves using the method of grouping and looking for relationships among the variables to simplify the solving process through integration.
Tips
- Not correctly identifying the values of p, q, and r: Ensure that you interpret the equation's structure accurately.
- Mismanaging the separation of variables: Be cautious when cross-multiplying to avoid algebraic mistakes.
- Neglecting the constant of integration: Always remember to add the constant after integrating.
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