Question Statement

Find the general solution to the differential equation:

$\frac{dy}{dx}=\frac{y^3}{x^2}$

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Background and Explanation

This is a first-order separable differential equation. To solve it, we rearrange the equation to isolate all terms containing $y$ on one side and all terms containing $x$ on the other, then integrate both sides with respect to their variables.

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Solution

Step 1: Separate the variables

To solve the equation, we first move the $y$ terms to the left side and the $x$ terms to the right side:

$\begin{array}{rl}\frac{dy}{dx}& =\frac{y^3}{x^2}\\ dy& =\frac{y^3}{x^2}dx\\ \frac{dy}{y^3}& =\frac{1}{x^2}dx\end{array}$

Step 2: Integrate both sides

Next, we integrate both sides of the equation. To make the integration easier, we rewrite the fractions using negative exponents:

$\begin{array}{rl}\int \frac{dy}{y^3}& =\int \frac{1}{x^2}dx\\ \int y^{-3}dy& =\int x^{-2}dx\end{array}$

Applying the power rule for integration, $\int u^{n}du=\frac{u^{n+1}}{n+1}+C$:

$\begin{array}{rl}\frac{y^{-3+1}}{-3+1}& =\frac{x^{-2+1}}{-2+1}+c_1\\ \frac{y^{-2}}{-2}& =\frac{x^{-1}}{-1}+c_1\end{array}$

Step 3: Simplify the expression

We simplify the fractions to make the equation more readable:

$\frac{1}{-2y^2}=\frac{-1}{x}+c_1$

To isolate the $y$ term, we multiply the entire equation by $-2$:

$\begin{array}{rl}\frac{1}{y^2}& =\frac{2}{x}-2c_1\end{array}$

Finally, since $c_1$ is an arbitrary constant, we can replace the term $-2c_1$ with a single constant $c$ to represent the general solution:

$\frac{1}{y^2}=\frac{2}{x}+c(\text{where }c=-2c_1)$

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Key Formulas or Methods Used

• Separation of Variables: A method used to solve differential equations by grouping variables on opposite sides of the equals sign.
• Power Rule for Integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ (for $n\ne -1$).
• Constant Manipulation: Combining multiple constants or coefficients into a single arbitrary constant $c$.

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Summary of Steps

1. Rearrange the differential equation to separate the $y$ and $x$ variables.
2. Rewrite the terms as powers (e.g., $y^{-3}$) to prepare for integration.
3. Integrate both sides using the power rule and add a constant of integration $c_1$.
4. Multiply by $-2$ to simplify the coefficients.
5. Define a new constant $c$ to provide the final general solution.